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$\sin(x-y)=\sin x \cos y - \sin y \cos x$ after that and series expanding for $\frac{1}{x-y}$ the double integral is dividing into two separate in x and in y.
A possible plan to attack. 1. Divide domain of integration on $x<y$&$x>y$. 2. Take $\frac{1}{x-y}$ as series in these domains. 3. use $\sin(x-y)$. 4. Integrate two one-dimensional integrals in series. At least you will have an answer as some series.
There is a very good book of N.V.Kuznetsov Trace formulas and applications in analytical number theory-but it is in Russian. Note that a generalization of Selberg formula is called The Kuznetsov Trace formula (name due to Ivaniec, Huxley and Sarnak).Book of P.Sarnak Some applications of modular forms-the classical one.
Note also a chapter 11 in the book Poularikas, A. D. (Ed.). The Transforms and Applications Handbook. Boca Raton, FL: CRC Press, 1995. It contains a section on Discreet Mellin Tr.
