Hello Sol! Because "regret" and "remorse" are both feelings or emotional states I would say the following: In this context the correct phrasal verb to use (when speaking about regret and remorse) is "hold onto" or the verb "harbor". So the correct way to say it is: You shouldn't "hold onto" regret. / You shouldn't "harbor" regret. or You shouldn't "hold onto" any regrets. / You shouldn't "harbor" any regrets. and You shouldn't "hold onto" any remorse. / You shouldn't "harbor" any remorse. I hope that helps!

It would be better to use the verb have rather than hold. You shouldn't have any regrets. You shouldn't have any remorse.