The most common and most likely answer to this crossword clue is the 9 plus 4 letter word ATHLETE'S FOOT
A
T
H
L
E
T
E
'
S
F
O
O
T
Buddy explains!
The straight part
We know that "common complaint with trilbies" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case common complaint with trilbies = athlete's foot.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.