The most useful of all presentation casts is a Collapsed Cast. This is where the leader fails to completely straighten and the lands in a crumpled heap. This is very easy to perform using a Roll Cast, simply by doming the stroke, quite hard to make using an overhead cast and the most consistent way - for me anyway - is to alter trajectory so the flies collide with the water giving the same final layout configuration. This cast is absolutely essential for nymphing downstream and across tactics.
Possibly next of importance is the Reach Cast. The IFFF have decided to rename the Reach Cast and call it the Reach Mend. Which is a bit weird! I understand why of course, because it is technically a mend, but unfortunately instead of just saying that the Reach Cast is really a Mend and being done with it, they've actually decided to change the name. OK, so now you have a Reach Mend - where the cast is made and the reach follows as a mend, and the Reach Cast where the cast is made horizontally and the rod simply left at an angle - technically this is a Reach Cast! Anyway, it's a very useful cast, particularly when casting beyond a current (i.e. reaching upstream). But it can also be used to place the fly close to you, if a fish rises there, or can be used to set up the retrieve so that the fly doesn't swim directly towards the boat, can be used to set up a fly swimming along a bank or a weed bed and so on.
Similar to this is the Curve Mend. Which achieves a very similar thing however there are some differences. Obviously when fishing a streamer the Reach Cast/Mend is best because you keep a straight line. For dry fly you can use either. Where the curve mend is most useful for me is for placing the line around objects (and no so much for overcoming currents).
The Hump Mend on the other hand is quite useful. Particularly when you need to dump some slack into a complicated current mess, or indeed at the end of the fly line. The Hump Mend takes some practise. It's more of a dry fly cast for me. Nymphs I will collapse as preference (because it's both more consistent and the nymphs sometimes make a collision or land with a crash).
Then we have overpowered and underpowered curve casts. Overpowering is very useful, both for casting in and around objects, but also for downstream dry fly presentations, which is even more useful if the hook is followed up with one or more stack mends. Underpowering is much less useful for me, you can get consistent by changing trajectories, but at the end of the day I personally rarely use this one. Overpowered however is very useful and worth the effort to learn. IMO the Overpowered backhand cast is the most difficult of all these casts which have significant use.
There are a few other popular casts, such as the wiggle cast and the pile cast. I don't find either of these particularly useful when fishing. They are both downstream presentations, they each have their place but to be honest I only use the pile cast a couple of times a year and the wiggle cast (mend!) hardly ever.
A cast that I often use is a slightly inclined forward delivery which lands as a crumple in the leader and some fly line end. This is a highly useful cast. It is one of the types of Puddle Cast.
There are lots of other casts and mends but these are the most useful for me. One that I haven't listed here is the Tailing Loop collapse, which I use as an alternative to the Tuck Cast for sinking and collapsing heavy flies. I do use the Tuck Cast too, but rarely and it's more for banging a fly down such as a grasshopper in my hands.
Have a great week - sorry for the late FP!