Yes, they all have about the same meaning, but in AmE you also 'go' crazy/insane, versus 'become'.
They're all commonly used, but according to this Ngram, the order of frequency is crazy, mad, nuts, an insane in decreasing order of use.
Insane has a slightly more clinical sound to it. Crazy is also widely used as something more of a synonym for 'unbelievable' or 'chaotic', without the strict sense on insanity.
Really, John's finally getting married? That's crazy, I thought he'd never settle down.
The grocery story was crazy this morning, it was so crowded.
To a lesser extent, I sometimes hear it used as a synonym for something like 'very' or 'incredibly'.
It was crazy hot this afternoon, it hit 104.
Current slang may still be using 'cray cray', derived from crazy, meaning again 'really crazy', often though in a sarcastic sense.
There was a fad for a while (not sure if it's still so popular) to abbreviate words, so you'd hear things like 'cray' and 'evs (ever), as in like, "what evs, I'm out."
Edit. Also, you can use 'go nuts' to mean something like do a task/job in a frenzied manner, or to go overboard (to an extreme) in doing something.
Wow, Jane really went nuts on the party decorations, didn't she?
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