What situations you would encounter
Convert to single speed because of chain or derailleur issues
Adjust a seat
Unlikely, but possible - spoke or wheel out of true
What to look for in a tool
The bits and tools you would need
What to avoid in a tool
size - I guess it is nice to have extra tools, but if it is too big, you are less likely to be taking it somewhere
Make sure you can actually use the tools. If the tool body is too weak to actually turn an allen key. Another problem in some tools it that they are too compact. If too compact it can be hard to fit the allen key in.
The best tool for most people is a crankbrothers m17, crankbrothers m19, or a topeak mini pro 20
Something like the crank brothers M17 or M19 gets recommended a lot
Topeak tools are also highly recommended, for example the Topeak mini pro 20.
The topeak alien is recommended a ton because it was one of the original multitools. It is one the do everything approach.
Alternative approach: a small, modular kit
The approach I favor is to build out a small kit of tools.
I've spoken before about why I prefer modularity in my gear.
For modular tool kit:
some sort of multi tool - you need allen keys, torx bits for lots of mountain bike use, is there a tool that lets you use actual bits and still is small, I guess a set of l-shaped allen keys still is small, and a T25 torx. wega seems to be a great brand
small box wrench tool
a small knife - i would go with the leatherman micra, it has a blade
Put all of that in a saddle bag