divorce...where to file?

So who informs the Home Office you're divorced so the visa can be cancelled? In most cases the Family Court does not even inquire about a person's nationality or visa status.

My guess is if you went to the dr.'s, or paying taxes in from your income, might get you found out.  But that's totally a random guess.  I would imagine it definitely happens if you leave and try to re-enter.  Plus if there is bitterness after a divorce, one party could always turn someone in. 

Ok I've been thinking about this a bit more.  The FLR is sponsored by the British spouse, hence it follows that if the couple split up, the sponsorship then fails and that is how the visa is invalidated.

Yes, Perspehone, as soon as I posted that I thought to myself a nasty ex would ring the Home Office!

Also, NB there are special provisions for cases where the relationship breakdown is due to domestic violence perpetrated by the sponsoring partner.

Does it make a difference if you  got a spousal visa in the US or an FLR (after a fiancee visa) in the UK?

I don't see why it would make a difference.  They are both dependant on the applicant and sponsor being married and living as a married couple.