In addition, both have no regrets and have no regret are possible. The former refers to more than one regret, whereas the latter refers to a specific regret:
1. Robert Brault quote - Today I bent the truth to be kind, and I have no regret, for I am far surer of what is kind than I am of what is true.
2. I have no regret leaving law for shoe making.
3. Normally I would not reveal so much about a film in case readers may be planning to see it, but in this case I have no regret in doing so.
In short, the noun regret is countable; e.g., I have one regret; I have many regrets.
Does that help?