Muon spin rotation and relaxation are mostly performed with positive muons.  They are well suited to the study of magnetic fields at the atomic scale inside matter, such as those produced by various kinds of magnetism and/or superconductivity encountered in compounds occurring in nature or artificially produced by modern material science.
The London penetration depth is one of the most important parameters characterizing a superconductor because its inverse square provides a measure of the density n of Cooper pairs.  The dependence of n on temperature and magnetic field directly indicates the symmetry of the superconducting gap.  Muon spin spectroscopy provides a way to measure the penetration depth, and so has been used to study high-temperature cuprate superconductors since their discovery in 1986.
Other important fields of application of µSR exploit the fact that positive muons capture electrons to form muonium atoms which behave chemically as light isotopes of the hydrogen atom.  This allows investigation of the largest known kinetic isotope effect in some of the simplest types of chemical reactions, as well as the early stages of formation of radicals in organic chemicals.  Muonium is also studied as an analogue of hydrogen in semiconductors, where hydrogen is one of the most ubiquitous impurities.