I've heard a rumor that Curiosity rover sent a recording about Phobos covering the Sun on the Martian sky. Did it happen? If yes, how frequently could such things be recorded?
No, because Mars can't have eclipses. Strictly speaking, Mars has only transits. The difference is that Mars's moons are smaller than the Sun as viewed from Mars, thus they don't block out the entire sun. Eclipses are defined as only occurring if the entire sun is blocked, or at least the vast majority. Phobos blocks out only about 60% of the sun at most. This could arguably be called a partial solar eclipse (But definitions are fluid, so...)
And yes, there have been several transits viewed by the rovers Curiosity and Opportunity and photographed. Here's pictures of some of them.