For an introduction to Leopoldt's conjecture, see part 3, Chapter X of "Cohomology of Number Fields", which is freely available here.

Write Leo($K$,$p$) for Leopoldt's conjecture for the number field $K$ at the prime $p$. If $K$ is an abelian extension of either $\mathbb{Q}$ or an imaginary quadratic field, then we know Leo($K$,$p$) for all primes $p$. (This is due to Ax and Brumer - see reference above for more details.)

Now let $K$ be a cubic extension of $\mathbb{Q}$. I am interested in Leo($K$,3). If $K/\mathbb{Q}$ is Galois then we are done. If $K$ has mixed signature, then its Galois closure $L$ is a totally complex $S_3$ extension of $\mathbb{Q}$ and thus is a cyclic extension of an imaginary quadratic field. Thus we know Leo($L$,3) and this implies Leo($K$,3) (Leopoldt's conjecture is inherited by subfields).

My question is: are there any cases for which Leo($K$,3) is known when $K$ is a totally real non-Galois cubic extension of $\mathbb{Q}$? By the above reasoning, it would suffice to find a totally real $S_3$ extension $F/\mathbb{Q}$ for which Leo($F$,3) is known.