I have long though about this myself.
RNA molecules are short lived everywhere you find them. There are enzymes always trying to break them up, even on your skin, as a protective measure.
If there are any RNA free in the nucleus, then part of the DNA is actively in transcription for them to maintain the population. The amount produced will vary depending on the mechanism that produces them. Some RNA molecules go on to modify other RNA molecules, and there is a "splicosome" in the nucleus that also participates in the enzymatic modification of RNA molecules, it may work with protein or RNA or both to change the mRNA message before it leaves the nucleus, and some never leave as there are ribosomes in the nucleus for producing those proteins that need to stay there.
It is very difficult to distinguish how and what the RNA is doing in the nucleus at any given instant. I know of microarrays and other technology that can be used to fish out sequences, but the whole process is convoluted you could not possibly ask any random person, "hey, how many non RNA are there in a given random liver cell, and what are their functions regarding the modification of mRNA in that cell?". The answer will vary cell to cell, job to job, and with environmental factors and age and more.
My area of interest is viruses. One day I hope to see a synthetic virus that infects a human cell, inserts a sequence to the DNA of that cell in a very particular place, and actively transcribes a sequence. That sequence is of course made of disposable RNA and then pairs with other viral RNAs, disabling or cleaving it. That to me is a better cure for a papilloma virus infection than the old fashioned method of killing the infected cells with liquid nitrogen. This would be one example of the use of non mRNA In a nucleus, as it is also used by our original cells to produce sequences that do the same thing (so called si-RNA and mi-RNAs can silence or disrupt their targets).
