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This might not be correct---perhaps someone can confirm---but I was once told (when I was a graduate student) that the way that Leopoldt's conjecture was proved for abelian number fields was as follows: first do the standard reduction to show that Leopoldt is true if certain special values of certain $p$-adic $L$-functions $L(1,\chi)$ are non-zero, and then prove that these numbers are non-zero by showing that they are transcendental! As I say, I don't know for sure if this is true, but my source was pretty reliable. The emphasis was on the observation that (at the time at least), apparently the only way of proving the numbers were non-zero was by showing they were transcendental.