The most common and most likely answer to this crossword clue is the 3 letter word RUE
I've seen this crossword clue in The Times
R
U
E
More clues leading to the result RUE
- Little friend, we hear, to bear regret
- Regret trick, sending out son
- Regret being impolite, putting daughter out
- Regret the French urban way
- Regret being endlessly cruel
- Feel bad about street in Paris
- Wish undone in French way
- French way to show regret
- Regret game ending in melee
- Mourn grouse after odd parts plucked
Buddy explains!

The straight part
We know that "to experience regret" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to experience regret = rue.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does RUE mean?
RUE noun- Sorrow; repentance; regret.
- Pity; compassion.
- To cause to repent of sin or regret some past action.
- To cause to feel sorrow or pity.
- To repent of or regret (some past action or event); to wish that a past action or event had not taken place.
- To feel compassion or pity.
- To feel sorrow or regret.
- Any of various perennial shrubs of the genus Ruta, especially the herb Ruta graveolens (common rue), formerly used in medicines.