The MEG experiment, named after "muons", "electrons", and "gamma", tries to observe muons that decay to an electron and a photon. It's been initiated in Japan but it is currently running in Switzerland.
μ- → e- + γ or μ+ → e+ + γSo far, no reliable evidence of such a flavor-changing decay has been observed. That's also true for the MEG experiment as of May 2010. The upper bound on the branching ratio is something like "at most 10-30 parts per trillion" among the decays of muons end up with electrons and photons.
Experimentally accessible decays of the muon are predicted by supersymmetry - and perhaps other beyond-the-Standard-Model physics scenarios. The detailed fraction of the decays with this fate depends on unknown values of various parameters.
However, the Symmetry Breaking Magazine just published something that could be interpreted as a positive rumor about a signal:
America's Fermilab will start the construction of its own muon-to-electron experiment, Mu2e experiment, in 2013 - in the post-Tevatron era. But the data over there will only be collected around 2017.