Not just any day out in public, no, I don't think it's appropriate for children to dress like a music artist - but it can be fun allowing your child to dress up like their favorite singing sensation for a party, and even help them with their "look." If the party is not at your own home, and you have to travel to the party, stopping at a store along the way or on the way back, and go into the store with your child, then I guess that would be in public, but only temporarily.
I include a photo of my son taken two years ago when he was 12. It was the end of his primary section of school, and this is how he dressed to attend the celebration function - it was themed 80's stars, and here he is dressed as Billy Idol.
But simply allowing young children to dress as they want to is not something I believe is "right." Some guidance should definitely be given, and consequences of dressing innapropriately gently and carefully explained.
To humor a child who wants to dress like their favorite star, throw a party for that child, preferably at your own home, and definitely with a few responsible adults there too - all the children attending can dress like their favorite stars. For one night only (until a friend of theirs throws a similar party!)