The most common and most likely answer to this crossword clue is the 4 letter word FAME
I've seen this crossword clue in Independent
F
A
M
E
More clues leading to the result FAME
- Notes about Mike’s celebrity status
- A female making comeback, this person is celebrity
- Celebrity's unavailability — not at home
- Following a number one celebrity
- Celebrity, female wanting the last word getting cut short
- Celebrity writing note to self?
- With nothing I make celebrity status
- Lover left out celebrity
- Prestige of footballers and yours truly!
- Celebrity in spoof, a megastar
Buddy explains!

The straight part
We know that "celebrity" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case celebrity = fame.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 FAME mean?
FAME noun- What is said or reported; gossip, rumour.
- One's reputation.
- The state of being famous or well-known and spoken of.
- To make (someone or something) famous