Surely there's nothing to cleaning mouthpieces - a drop of detergent in some lukewarm water (or cold water if it's a vintage ebonite mouthpiece - hot water will turn some ebonite mouthpieces green, and the older the piece is the more likely it is to be susceptible), a bit of gentle scrub with a soft mouthpiece brush - how difficult can it be?
From the state of some of the mouthpieces I see I reckon some people must find it incredibly hard.
But it's not only water that gets blown into the mouthpiece; there are oils, fats, salts, starches, sugars etc. - and some of these can shrug off the effects of your usual cleaning regime.

Calcium (or more correctly, Calcium Carbonate) is one the worst offenders - small deposits attach themselves to the walls of the bore, and little by little, day by day, they build up. It's exactly the same stuff that forms in the bottom of your kettle, otherwise known as scale (or limescale, to give it its full name).
Obviously, regular cleaning can slow this build-up process but even the most fastidious player will suffer from some degree of scale build up. More often than not the problem is most acute when buying secondhand mouthpieces. You may well clean your mouthpiece out on a regular basis, but that doesn't mean everyone else does.

Have a look at this!

This is a Boosey & Hawkes 926 clarinet mouthpiece - it's at least 40 years old, and looks like it was played every single day. Someone loved it enough to repair it when the tip broke off (though they didn't love it enough to take it to someone who could do a proper repair).
See that white stuff inside on the floor? That's the scale build-up. It adheres to the mouthpiece and refuses to be shifted even by fairly brisk scrubbing. Even worse, the more there is the more will form - and it provides an excellent surface for bacteria to breed on, plus it affects the response of the mouthpiece.

You can't attack the scale with an abrasive, or a sharp implement - it will only result in scoring the bore of the mouthpiece. What's really needed is a chemical that dissolves the scale and leaves the mouthpiece untouched - but which chemical?
There are plenty of products on the market for removing calcium carbonate deposits - but very few of these products are suitable for use on mouthpieces and many of them will ruin the mouthpiece altogether. Most of these limescale removing products are based on acids - limescale is an alkaline substance, acid will dissolve it.
What's required is an acid that's relatively mild in action, that will attack the scale but not the mouthpiece.

There are two chemicals commonly touted as being suitable for the task - and both are readily available to the general public; domestic Hydrogen Peroxide Solution (9%) and vinegar. Hydrogen Peroxide is available from your pharmacist - it's used as a mild disinfectant for skin wounds...if you have your ears pierced this is the stuff you put on the wound to prevent it becoming infected.
I'm going to try both and see what happens. I'm also going to try a few other products that I've seen recommended by various players to see how well they perform in comparison.

I placed the mouthpiece above in a jar containing Hydrogen Peroxide. It immediately began to gently fizz, which looked promising. It carried on fizzing for quite some considerable time - so much so that I got bored of watching it and decided to carry on with the second test.

Here's a Selmer S80 D alto saxophone mouthpiece. There's about the same amount of scale build-up as in the clarinet mouthpiece - it looks less as it's spread out over a larger surface area, and there's a bit in the bore than can't be seen in the photo.

And here's the same mouthpiece after 30 minutes immersed in white malt vinegar.

You can quite clearly see that the scale deposits have gone. What you can also see is that the mouthpiece has taken on a dull greenish hue. This is due to the action of the vinegar on the Ebonite that the mouthpiece is constructed of.
This does no great harm to the mouthpiece, but doesn't exactly look terribly attractive. It's possible to remove this 'bloom', but it's a job best left to a repairer as it requires the use of abrasive compounds.

From my observation of the action of the vinegar it seems that it appears to soften up the scale rather faster than it dissolves it. Obviously you could wait until the vinegar dissolved the scale of its own accord - but by that time the mouthpiece would be a great deal greener - and I suspect that a degree of softening of the ebonite would take place, at least on the surface.

With this in mind, I repeated the experiment - but this time I concentrated the effects of the vinegar in the appropriate spot.

I've simply soaked a wad of cotton wool in the vinegar and placed it in the mouthpiece's window. I then propped the mouthpiece up so that the wad lay flat and the vinegar didn't dribble off into the bore and left it for ten minutes.
I then removed the wad, and with the aid of a cotton bud soaked in vinegar gently rubbed off all the scale deposits.
There was a very slight greening of the ebonite in the bore - but by no means as much as the previous experiment. I found that ten minutes was quite sufficient to clean this moderately scaled mouthpiece - more heavily encrusted examples would need another dose with the wad. Confining the vinegar to the bore meant that the exterior finish of the mouthpiece remained untouched.

Some two hours have passed since I placed the clarinet mouthpiece in the Hydrogen Peroxide. I took it out and examined it.
This chemical has a different action to the vinegar - instead of softening the scale it seems to slowly dissolve it. I wonder how much of this is due to the small amount of Phosphoric acid the product contains? Compared with the photo at the top of the page you can see a considerable reduction in the scale deposits. I tried rubbing the deposits off with a cotton bud, but to no avail - this chemical simply has to be left to do its work in its own time.
You may notice too that the ebonite hasn't really gone any greener than it already was. I did notice a very slight greening effect, but it wasn't anywhere near as much as seen on the vinegar treated piece.
Just out of curiosity I cleaned the SA80 piece and popped it into the Hydrogen Peroxide. After 15 minutes or so it had gone green again - so it appears that some types of ebonite are more susceptible to discolouration than others. It could also just be that ebonite tends to deteriorate over time depending on what environment it's been kept in - a test piece of new ebonite showed no discolouration after 60 minutes immersed in the vinegar.

I then tested a few other cleaning products.
The first of these was a kettle cleaner. This came in a small bottle, priced about £2, and contained a solution of 40% (w/v) formic acid.
The instructions required the solution to be tipped into a kettle with a cupful or so of boiling water. Clearly this isn't an option for an ebonite mouthpiece - so I used the solution neat and cold. And here's the result on the remaining deposits on the 926 mouthpiece after ten minutes.
Pretty much all the scale has gone - though it worked in much the same way as the vinegar, requiring a little gentle help from a cotton bud. There was no noticeable greening of the mouthpiece.
Much more noticeable though were the fumes given off - they were really quite unpleasant (there was a warning about this on the bottle).
I did the usual with the poor old SA80 mouthpiece - it went green again.

Baking soda has been suggested as a means of cleaning mouthpieces, which seems rather counterintuitive to me as it's an alkali (in fact it's less acidic than water) but I tested it anyway.
Here we can see a rather dirty Otto Link baritone sax mouthpiece with a modest coating of scale. A paste mix of water and baking soda was made up and then drizzled onto the scale and left for an hour. As you can see from the before and after shots there's hardly any difference at all, even with a saturated solution of the stuff. If there's any merit in using baking soda it probable lies in its action as a scouring agent when used with a small brush.

Another popular suggestion is cola. Cola contains phosphoric acid, and many other fizzy soft drinks contain citric acid.
You may well have tried that old trick of dropping a dirty old copper coin into a glass of cola overnight in order to clean it up - when you lift it out in the morning it's all bright and shiny (well, less dirty anyway).

These drinks should work in the same way as vinegar, though less vigorously as the concentration of acid is slightly weaker. It should be noted that, in general, diet fizzy soft drinks are less acidic than their full-sugar counterparts (and are about as acidic as red wine).
To see how effective cola is when compared to vinegar I rigged up a simple test. I've sealed up the bore of the mouthpiece with adhesive tack and divided the window down the middle. The left side was filled with fresh non-diet cola, the right with plain malt vinegar. The piece was left to soak for an hour at room temperature.

As you can see it's a clear win for the vinegar - the section on the right is completely clean, the section on the left is little better than the dividing line which was covered during the test and received no treatment at all.

For the final test I tried a dental product - denture cleaning tablets.
I bought the strongest tablets I could find - they claimed to remove heavy staining from dentures in 10 minutes.
I filled a small jar with hand-hot water, popped a cleaning tablet in, then added a mouthpiece.
Things fizzed for quite a while - but this turned out to be the tablet dissolving.
After ten minutes I examined the mouthpiece - no effect whatsoever.
I gave it another half hour. Nothing.
I gave it another hour - still no effect.
I'm still checking it even as I write this article several hours later. There's some brown scum on the surface of the solution, but the scale deposits are looking solid (if perhaps a little whiter!!?).

To sum up then: Hydrogen Peroxide, non-diet cola and vinegar did the job. Hydrogen Peroxide took by far the longest to achieve a clean mouthpiece - and because of this it's fair to assume that the whole mouthpiece would need to be immersed in the solution. A stronger solution might work faster...though I'm still inclined to think that it's the acid in the solution that's doing the work with regard to the scale.
Cola had a slightly better effect and acted faster, but plain malt vinegar beat everything else hands down.
Vinegar also acted fast to soften the deposits, and because of this it's possible to limit its contact with the mouthpiece by using a wad of cotton wool soaked in the solution to get things moving and then use a cotton bud (or Q-tip) to rub the scale off.
Proprietary limescale removers do the job - but the production of noxious fumes and their generally poisonous nature makes them risky and unpleasant to use.
The dental products and baking soda simply didn't work.

Another conclusion that can be drawn from these tests is that the well-meaning advice to 'dunk the mouthpiece' in various solutions is something of a risky proposition.
Plenty of people have reported no apparent harm coming to the mouthpiece when left in a cup of cola, but a great deal depends on what the mouthpiece is made of and what the state of the material is. Some ebonite seems more susceptible to greening than others, and prolonged exposure to light and heat tends to exacerbate this tendency (so older pieces are more likely to suffer). My advice is to play it safe - don't dunk - take the time and trouble to used soaked wads that can be placed in the mouthpiece's window unless you're absolutely and 100% sure that your mouthpiece will not be affected by complete immersion (such as a basic Yamaha mouthpiece, which is made of a hard plastic).

That said, there is one advantage of complete immersion, and that is that it will ensure complete coverage of the bore. This is important, as there can be just as much - if not more - scale build-up in the areas you can't easily see. As you can see here, the scale deposits in the window are scattered around...but just inside the bore they're starting to form a more solid coating. The simplest solution is to poke some cotton wool down the bore before you fill the window with it, and this will draw your chosen solution right inside the mouthpiece.

Although I used white malt vinegar I can see no reason why some other sorts of vinegar would not work just as well - such as common brown malt vinegar, as found on your chips (or fries, depending on where in the world you are) or cider vinegar. If in any doubt look on the label for the acidity - it will usually be in the region of 4% to 6%. Balsamic vinegar is not suitable.
As for metal mouthpieces, I did try a couple in both the Hydrogen Peroxide and the vinegar. Obviously colour change is not an issue - but metals tend to react adversely to acids, and with this in mind the speed with which the vinegar softens the scale makes it a viable proposition.

If you're more concerned with sterilising your mouthpiece then I would recommend washing it in lukewarm water with detergent, and if you're feeling particularly fragile you can give it a sloosh with some antibacterial mouthwash. Rinse off after a minute or so. (This stuff is good for slooshing out sax crooks too, which can get get just as gunked up as mouthpieces.
I've used some of those antibacterial surface cleaners with success (Dettox is a good one) - though care should be taken to avoid the ones that contain bleach or 'grease-busting' agents...they may be rather harsh. There's no need to be any more fastidious than that - the risks of catching something nasty from a mouthpiece are the same as those in sharing a cup, just use your common sense.
If you're still worried, think about what's on all those coins in your pocket...and have a read of this article.