I got a counter offer when leaving my last role, I did feel a bit offended by it as that organisation had not listened to any of my concerns, and after saying I was leaving for work / life balance, paying me more to continue with 14 hour days was not an option I wanted from them. It just re-inforced my opinion that the management had no idea of being able to listen to their employees.
I had a counter offer of a new role (and more money with it) in my first job I left, I did think long and hard about that one and didn't feel insulted at all (I was leaving to further my career)
In my
HR experience, I believe that most counter offers that only include money and have been accepted, have ended up in the employee getting a larger payout within 6 months when they have left anyway.