Perhaps the counteroffer is a guise as they hire somebody to replace you, then you'll be let go as soon as they hire somebody.
The relationship can never be as "warm and fuzzy" as before you tried to quit.
Two people are dating, one is an alcoholic. The other says, "I'm leaving, we've talked about your drinking 100 times and nothing changes so I'm leaving you. Bye." The drunk says, "oh no, stay with me and I'll stop drinking!"
Does that situation ever turn out well?
A person with a solid job that isn't on the chopping block (presumed because of the presence of a counteroffer) isn't job-shopping for a 10% bump.
Not so fast. I, for example never changed jobs for any other reason.
Of course, I never got only a 10% bump, but well, we do need money, and we should try to get more of it with time.
I would say never accept a counter offer. If you decided to leave there must be other things that bother you besides money.
Those reasons aren't going to change. So if you accept more money you would be happier for about 6 months and then reality will settle in... and would still be unhappy and planning to leave anyway.
Even if you are happy with your current job and the only reason to leave is to have a pay increase, when you receive a counter offer with more money you shouldn't stay either. Because a raise by means of counter offer, means that salary at your current employer aren't fair and are based just on negotiation skills alone.