БЫТЬ -to be
Where were you last Saturday?-Где ты был в прошлую субботу?
I will be there next Monday.-Я буду там в следующий понедельник.
БЫВАТЬ-to go somewhere often, regularly or sometimes, to be sometimes
She goes to the club at the weekend. -По выходным она бывает в клубе.
He is rude simetimes.-Он бывает грубым. (He isn't always rude.)
БЫВАТЬ-to happen
For example, somebody is telling their friend ," I was late for work yesterday and my boss was angry."
His/her friend is calming him/her down, "Don't worry, these things happen."-Не переживай, бывает.
I would say that in most cases the synonyms "случаться" and "происходить" mean "to happen".
How often does this happen?-Как часто это происходит? (That means that something happens quite often.)
"Cлучаться" comes from the word " случай" ("case","by accident"). Something happens suddenly or occasionally.-Что-то случается.
How often does this happen to you?-И часто с вами такое случается? (It means that something happens, but not very often.)