Point screws allow for keys to be securely mounted to the body of the instrument whilst still allowing them to move in an appropriate fashion. Typically you'll find them used on long single keys where a rod screw would be impractical - although sometimes it appears that the use of particular types of pivot screw is an entirely arbitrary decision.

There are three main types of point screw in common use:

From left to right we have the shouldered or proper point screw, the shoulderless point screw - and below that the parallel or cylindrical point screw.

The big advantage of the point screw is that it is constantly adjustable to take up the wear in the keys.
Note how the taper of the proper point screw butts up against the wall of a hole drilled in the key barrel. It follows that if there is play here then the key will wobble - and if the screw is too tight then the key will jam.
Note too the action of the cylindrical point screw. Not having a tapered point the best this kind of screw can do is simply hold the key in-situ - there's no facility for tightening the action other than having the ends of the key barrels swedged.
Regrettably this kind of screw is found on more and more instruments these days, particularly the student models, and if your instrument is fitted with these screws you'd be well advised to have them replaced with proper point screws at the next major service.

A variation on this theme is a screw with a point on its tip - a pseudo point screw, also known as the bullet-headed point screw.

The pseudo point screw bears closer examination.
Because it has a point on its tip it can act like a proper point screw in certain circumstances.

The diagram on the left shows how the screw can function like a proper point - its tip is clearly in contact with the key barrel, and there's a gap between the end of the barrel and the pillar.
This means that as the key barrel wears, the screw can be tightened further into the pillar and thus take up the wear.
However, in most situations where these screws are used the hole in the key barrel is deeper than the length of the screw's tip...so the point sits in air and can only function as a cylindrical point screw.

The tricky bit is that in order to find out if this is the case you have to remove a screw and a key and poke the screw tip into the hole in the key barrel to see if the tip meets any metal.

One place you're certain to find the pseudo point screw is on a modern Selmer saxophone. These instruments use a unique method to automatically adjust for any wear in the action by means of a sprung cylinder inserted into the bore of the key barrel.

As the tip of the screw wears away the contact point in the cylinder, the spring pushes the cylinder forward to close the gap ( obviously this wear happens at an extremely slow rate ). This ensures that the business end of the pseudo point screw is always in contact with the key barrel - and this means that you should never need to ream the pillars to take up wear in the action.

Sounds ideal, but there are one or two drawbacks.

Because the system requires that there's a bit of a gap between the end of the key barrel and pillar ( otherwise the sprung cylinder would simply butt up against the pillar and you'd be back to square one once the point screw tip starts to wear the barrel ), and because there's a spring in the barrel...or one at each end actually...you can never adjust the action so that it's tight.
By tight I don't mean stiff...but if you've ever wiggled a key properly fitted between two plain point screws you'll know exactly what I mean. The key can move up and down freely but there's no lateral movement.

The other drawback is that it's possible for the cylinder to wear in the key barrel...and then you're in trouble.

Technically speaking you can replace the cylinder...but if the barrel's worn then there's little point. It could get very messy!

Shouldered point screws are tightened by means of reaming out the pillars they sit in, thus allowing more of the point to extend beyond the pillar. Shoulderless points may be merely tightened further in - though it is necessary to lock them in place with a mild threadlocking solution afterwards to prevent them working loose ( or tighter! ).
Some instruments feature such screws with nylons collars backing onto the head of the screw. This acts as a built-in locking device.