Someone once said to me: "Muscles use energy even though they aren't working." and I was blown away. Nevertheless this isn't the most thrilling question of all times, at that time it was to me. Our system is much more convoluted than we might first think. And that's more or less why we are studying it around-the-clock. It's definitely not simple. Vaccines and medicines can have surprising side effects when our system responds in an unexpected way, as we have seen for example with the much speculated H1N1 vaccine in Finland and other countries possibly causing narcolepsy.
What comes to the muscles using energy while relaxed, I pondered that it might have something to do with the calcium pumps in the muscle cells that pump the Ca++-ions actively back to the sarcoplasmic reticulum after a muscle contraction, constantly keeping the Ca++-concentration low in the intracellular fluid. I'm sure there are many other things too that use energy but this was one thing that crossed my mind.
Regarding to muscle building and medical science I found a nice chapther from Galenos. This chapter tells how creatine phosphate works as a phosphate buffer. At the beginning of our muscular effort we use mainly the energy gotten from ATP (adenosine triphosphate) forming to ADP (adenosine diphosphate). Creatine phosphate then anaerobically donates its phosphate group to ADP to form ATP again and thus we can get the same energy again by ATP forming to ADP. Quite logical but it was cool to find it from the big G too.