Let \(p\) be an odd prime number and \(G\) the non-abelian group of order \(p^3\) and exponent \(p\). The author shows there exists Galois extensions \(L\) of \({\mathbb Q}\) with group \(G\) and vanishing Iwasawa invariants \(\lambda,\mu\) and \(\nu\) for the cyclotomic \({\mathbb Z}_p\)-extension of \(L\). He has also a result for extensions \(L/k\) where \(k\) is an imaginary quadratic field with \(p\) a prime not dividing the class number of \(k \) and furthermore unramified in \(k\) if \(p=3\); he proves the existence of such extensions \(L/k\) verifying Leopoldt's conjecture. For this he uses an old criterion found by \textit{H. Miki} [J. Number Theory 26, 117-128 (1987; Zbl 0621.12009)].