これは英英辞典の出番ですね。私の辞書では４つが挙っていました。日本人が信じている？ニュアンスとは微妙なズレもありますので、要注意です。（日本語の残念、ごめんより sorry, regret はもう少し広いニュアンスで使われる場合があります）
sorry = used for emphasizing how bad, stupid, or embarrassing something is
-----That's the sorriest excuse I've ever heard.
sorry = ashamed, embarrassed, or unhappy about something that you have done
-----Tell your sister you're sorry!
sorry = feeling sadness or sympathy for someone because something bad has happened to them
-----I'm sorry about your losing your job.
sorry = disappointed about a situation, and wishing you could change it
-----I'm sorry you have decided to leave the company.
regret は動詞 (or 名詞) ですので sorry (形容詞) とは品詞が違いますが、意味的にはよく似ています。
regret = to feel sorry or sad that something has happened (or, to feel sorry or sad about something that you have said or done)
-----We regret any inconvenience caused by the delay.
-----Haven't you ever said anything you regret?
