I regret I didn't do it.
I regret I didn't do that.
I regret not doing it.

Are all understandable, have the same meaning, and are used by native speakers to say they regret not doing something.

I suggested (that) I should do it (instead of my brother).
Thanks for reminding me, I forgot I (already) did that.
I explained (to him that) I did it, and not the other guy.

are other phrases which might be used in informal conversation.

I will say that "I regret I didn't do it" sounds more AmE than BrE. AmE speakers tend to use "I" much more that BrE speakers from my observations. For example, the Queen will famously use the royal "we".

I regret I didn't do it.
I regret I didn't do that.
I regret not doing it.

Are all understandable, have the same meaning, and are used by native speakers to say they regret not doing something.

I suggested (that) I should do it (instead of my brother).
Thanks for reminding me, I forgot I (already) did that.
I explained (to him that) I did it (instead of the other guy).

are other phrases which might be used in informal conversation.

I will say that "I regret I didn't do it" sounds more AmE than BrE. AmE speakers tend to use "I" much more that BrE speakers from my observations. For example, the Queen will famously use the royal "we".