Silicon photomultipliers have received a great deal of interest recently for use in applications spanning a wide variety of fields, including nuclear safeguards and nonproliferation. For nuclear-related applications, the ability of silicon photomultipliers to discriminate neutrons from gamma rays using pulse shape discrimination when coupled with certain organic scintillators is a characteristic of utmost importance. This work reports on progress characterizing the performance of twenty different silicon photomultipliers from five manufacturers with an emphasis on pulse shape discrimination performance and timing. Results are presented on pulse shape discrimination performance as a function of overvoltage for 6-mm x 6-mm silicon photomultipliers, and the time response to stilbene is characterized for silicon photomultipliers of three different sizes. Finally, comparison with a photomultiplier tube shows that some new-generation silicon photomultipliers can perform as well as photomultiplier tubes in neutron-gamma ray discrimination.