This paper is concerned with determining which quadratic forms over a field \(K\) are trace forms of \(G\)-Galois extensions \(L/K\). M. Epkenhans had earlier conjectured that in the case when \(K\) contains a primitive fourth root of unity, the trace form \(q_{L/K}\) is always a scaled Pfister form. A proof of this conjecture is one of the main results of this paper. In particular, if \(K\) is a field containing a primitive fourth root of unity, \(L/K\) is a \(G\)-Galois extension, \(S\) is a Sylow 2-subgroup of \(G\), and \(r\) is the rank of the elementary abelian group \(S/S^2\), then the trace form \(q_{L/K}\) is Witt-equivalent to the scaled Pfister form \(\langle | S| \rangle \otimes \langle\langle a_1, \dots, a_r\rangle\rangle\), for some \(a_1, \dots, a_r \in K^*\). The second main result of the paper is a complete description of those finite groups \(G\) which admit a \(G\)-Galois extension \(L/K\) having a nonhyperbolic trace form, here under the assumption that \(K\) contains a primitive root of unity of degree \(2^m\) for some fixed integer \(m \geq 2\). Finally, some applications of the two main theorems are provided, and a description of all quadratic forms that can occur as the trace form of a \(G\)-Galois extension for \(G \cong M(2^n) = \langle \sigma, \tau \mid \sigma^{2^{n-1}} = 1 = \tau^2, \tau\sigma\tau = \sigma^{1 + 2^{n-2}}\rangle\).