Giggle's beginning to interrupt dreadful sad song (5)
I believe the answer is:
dirge
'sad song' is the definition.
(I've seen this in another clue)
'giggle's beginning to interrupt dreadful' is the wordplay.
'beginning to' says to take the initial letters.
'interrupt' is an insertion indicator.
'dreadful' becomes 'dire' (synonyms).
The first letter of 'giggles' is 'g'.
'g' put into 'dire' is 'DIRGE'.
(Other definitions for dirge that I've seen before include "Sombre music" , "Funereal hymn" , "Funeral hymn" , "Elegy" , "No joy in this music" .)