The famous Leopoldt conjecture asserts that for any number field $F$ and any prime $p$, the $p$-adic regulator of $F$ is nonzero. This is known to be equivalent to the vanishing of $H^2(G_{F'/F},\mathbf{Q}_p/\mathbf{Z}_p)$, where $F'$ is the maximal pro-$p$ extension of $F$ unramified outside $p$. When $F/\mathbf{Q}$ is abelian, the conjecture was proven by Brumer.

My question: is there any reasonable sense in which the Leopoldt conjecture is "usually" true - e.g., is it known for any fixed $F$ at almost all primes $p$, or (say) for almost all quartic extensions of $\mathbf{Q}$ with $p$ fixed? A glance through the mathscinet reviews of all the papers with "Leopoldt conjecture" in their title didn't reveal anything, but perhaps this is well-known to experts.

The famous Leopoldt conjecture asserts that for any number field $F$ and any prime $p$, the $p$-adic regulator of $F$ is nonzero. This is known to be equivalent to the vanishing of $H^2(G_{F'/F},\mathbf{Q}_p/\mathbf{Z}_p)$, where $F'$ is the maximal pro-$p$ extension of $F$ unramified outside $p$. When $F/\mathbf{Q}$ is abelian, the conjecture was proven by Brumer.

My question: is there any reasonable sense in which the Leopoldt conjecture is "usually" true - e.g., is it known for any fixed $F$ at almost all primes $p$, or (say) for almost all quartic extensions of $\mathbf{Q}$ with $p$ fixed? A glance through the mathscinet reviews of all the papers with "Leopoldt conjecture" in their title didn't reveal anything, but perhaps this is well-known to experts.