Grammatically correct

I feel 'singing girl' works, but I would say 'the boy who is running.' I don't know why though. Let's see if anyone else knows why.

I find both of them similarly odd. I think a native speaker would always use the relative clause as English 1b3 has done:

Do you know the boy that's running over there?
Do you know the girl who's singing now?

Usually, post-modification is for temporary conditions and pre-modification is for more permanent traits:

Do you know the blond boy?

Hi,

Are these correct?
Do you know the running boy?
Do you know the singing girl?
The singing girl is Jana.
The running boy is Mark.

Yes, they are correct. But not very commonly said.

This form is more natural.
Q - Do you know the boy who is runnin

Would these be more natural if I leave out who or that?
Do you know the boy running?
Do you know the girl singing?
The girl singing is Jana.
The boy running is Mark.
Cheers Fulvio

yellowstarstruckWould these be more natural if I leave out who or that?

Not more so, but equally so, in the spoken language.