Giggle's beginning to interrupt dreadful sad song (5)
I believe the answer is:
dirge
'sad song' is the definition.
(I've seen this before)
'giggle's beginning to interrupt dreadful' is the wordplay.
'beginning to' suggests taking the first letters.
'interrupt' is an insertion indicator.
'dreadful' becomes 'dire' (synonyms).
The initial letter of 'giggles' is 'g'.
'g' going within '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" .)