How long to keep ASRS report?
Generally the FAA inspectors are supposed to complete their EIR within 75 days of the date they learn of the violation, and regional review should be completed 15 days after that - so you should theoretically hear something from them within 90 days of a violation. 

The NTSB's rules of practice governing certificate actions (referenced in that same FSIMS order - 2150.3B) says they can't suspend or revoke your certificate if the violation took place more than 6 months prior to their notifying you (the "stale complaint rule") so generally if you haven't heard from them after 6 months you're probably in the clear and won't need your ASRS stub anymore, but there are exceptions to this that are easy for the FAA to leverage, so holding on to the stubs longer isn't a bad idea.



Personally I believe in holding on to the ASRS stubs forever - I have mine (all 3) in a folder in the back of my filing cabinet, even though the newest one is almost 3 years old now and I'm fairly sure the FAA doesn't care about any of them, but if they do come after me for any of those incidents I know I did the right thing and I can prove it.  

More importantly for me though it's a reminder that the point of the ASRS program isn't to be a get-out-of-jail-free card for when we screw up: That's the carrot used to entice us into filing them, but these reports are supposed to serve as an educational tool for other pilots and for us.
Any time I feel the need to file another ASRS report I take the time to look at the ones I filed previously - If I'm making the same mistakes multiple times that's a clue that I should be getting time with a CFI and working on those areas of operation.

Knowing my insurance policy on those incidents is still good if someone dredges them up later is just a bonus, and the stubs don't take up that much room.