Regrets generally serve no purpose, but remorse can be useful if properly applied. Most people feel regret over the wrong things. They feel regret over circumstances outside of their control and outcomes that didn't go their way. That is a waste of emotional energy. However, most people also don't feel proper remorse over poor decisions they made chosing instead to rationalize or minimize their guilty feelings and push away the remorse that might prevent them from repeating those mistakes in the future.
When I was in 7th grade, I had a friend that I teased to the point he told me to go away. I lost a friend over my poor behavior. After that, whenever I thought it might be funny to tease someone, I remembered the negative outcome that came from doing that, and I stopped myself from continuing that negative behavior. That is the value of remorse.
I have several mistakes from my past that I keep alive because the remorse prevents me from repeating the mistake. I don't dwell on it or beat myself up for being a terrible person, but I listen to those feelings of remorse when it's appropriate to the circumstances. I think it has served me well.