For supersymmetry and supergravity, my primary recommendation is
Supergravity by Freedman and Van Proeyen. This book has a very large number of exercises interspersed across the text. The exercises are placed at locations that are relevant to the prose around them, and they vary in their level of difficulty from verifying certain results in the text, to involved problems, so they're well-suited to self-study. As an aside, the prose itself is (at least in the first 10 chapters or so) clear and pedgogical. Although this is not strictly a problem book, it has so many of them that it could effectively function as one for someone who treats it that way. This book will be useful if you have a relatively strong background in QFT and are just getting into research in theoretical high energy.
A secondary recommendation is sections 3.6 and 4.12 of the classic review
"Supersymmetric Gauge Theories and the AdS/CFT Correspondence" by D'Hoker and Freedman. These two sections contain five and four problems respectively on SYM and sugra/superstrings. Although there are not a large number of exercises here, the exercises are interesting and relevant. The level is suitable for anyone with a strong background in QFT and especially the mathematics of symmetry in physics (groups, algebras, etc.)
This post imported from StackExchange Physics at 2014-12-12 22:28 (UTC), posted by SE-user joshphysics