The most common and most likely answer to this crossword clue is the 4 letter word LILT
I've seen this crossword clue in The Daily Telegraph
L
I
L
T
More clues leading to the result LILT
- A jaunty rhythm inspired by the Caribbean as an alternative to 29?
- Song books, about fifty
- Sing happily till drunk
- Sing merrily being drunk, having consumed litres
- Plainly now and then with time speak with swinging rhythm
- Beat student taking a little time
- Lieutenant, appearing about middle of film, has a pleasing accent
Buddy explains!

The straight part
We know that "Sing happily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case sing happily = lilt.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 LILT mean?
LILT noun- Animated, brisk motion; spirited rhythm; sprightliness.
- A lively song or dance; a cheerful tune.
- A cheerful or melodious accent when speaking.
- To do something rhythmically, with animation and quickness, usually of music.
- To sing cheerfully, especially in Gaelic.
- To utter with spirit, animation, or gaiety; to sing with spirit and liveliness.