This is totally dependent on the design and also the materials you are using. EdStainless touched perfectly on the design. On the Materials side, if you are working with Metals/alloys/MMC, I would say tensile is more important. On the other hand, Ceramic/some PMC and CMC/powder metallurgy stuff, compression is the main parameter.
Of course tensile test would give you more data regarding the mechanical properties (yield, UTS, toughness, resilience, etc.)