And where do I start these wave events? In main menu I've got prologue, story and versus. I don't see any wave mode or something like that. Please point me to it so I can take a look on the crash.
Go to story mode -> Deadpool's Daily
then you'll have 5 nodes. The first one has 3 stars above it and is locked and there are 2 nodes just under this locked one: the one to the left is a common battle, the one to the right (called "Under the Sea" is a wave battle.
So I can use this for the single player missions without risk such as Deadpool daily and the non-alliance events they use?
You can use it everywhere, but keep in mind that winning against a team with Old Man Logan / Hulkbuster / Jean Gray with a 1*, 2* or 3* will probably get you reported. Just be reasonable and you'll be clean.