It's hard to list all of the sorts because sometimes the needs are quite nebulous. Sometimes it's someone who's a great engineer who's just... really great at certain kinds of growth hacking, whatever that entails. These are more "you know it when you see" opportunities. Engineering leadership can be sort of similar. The kind of person who can "make things happen". More technical roles are often easier to describe. A few concrete examples:

Someone who got their PhD researching databases who is also a very experienced software engineer.

A senior software engineer who's worked for a few years in the finance world, and understands some of the more advanced math like stochastic calculus.

Someone with experience helping hypergrowth mode startups whose main DB is MySql to adopt Vitess.