Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent <a href="http://www.sju.edu/~smith/pdf_files/OWR_2009_19.pdf">Oberwolfach report</a> from a meeting on the subject. 

If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.