It is singular. "In public" means in a place where anyone is free to come and go. To be picky - there are public places but there are also private places that one would still use the term "in public" just because there are other people around (other than family, friends...). True public places are streets, parks, public transportation, etc. Private places are shops, schools....

'In public' is an idiom. We use it when when other people, especially people you do not know, are present: 'It’s illegal to walk naked in public.' 'They kissed each other in public.' 'We must be careful everytime we are in public.'