DNA has transcribed into RNA (Non-coding). Can this RNA mutate and become a Protein-Coding one/mRNA? Have there been any such instances reported by scientists?
Possibly, but with such a low frequency as to be unimportant and undetectable. (Monkeys with typewriters producing Shakespeare’s Hamlet comes to mind.)
No. Because it would be extremely difficult to detect, it would seem to be of no importance if it did occur at a very low frequency (you make no suggestion of why it would be of interest) and therefore there would be no reason for anyone to try.
Why would it be difficult to detect?
The difference between mutations that occur in DNA and any that might occur in RNA, is that a mutation in DNA can give rise to a population (e.g. of bacteria, or offspring) all of which carry the mutation. Thus, even though the chance of the particular mutation may be very low (say 1 in a million or less), a situation can arise in which the whole population is producing the RNA transcribed from the mutated gene and the protein encoded by the RNA. If RNA mutation occurs with the same frequency and managed to change a non-coding RNA (perhaps a pseudogene with a frameshift) to a coding RNA, only 1 in a million transcripts would be changed. Therefore if one sequenced the RNA the incidence of such changed RNA would be too low to detect, or if detected would be below the error margin in such experimental work.
Why has nobody looked for it?
Experimental science costs time and money. Scientists explore theories that have an importance and probability of success that attracts competitive funding. If there is no particular reason to look for something and the chances of seeing it are minute, nobody will waste their time on it, and they would certainly not get funded to do so.