Let \(X\) be a topological vector space. A linear continuous operator \(T:X\to X\) is said to be hypercyclic if there is a vector \(x\in X\) (called hypercyclic vector for \(T\)) such that its orbit under \(T\) is dense in \(X\). A hypercyclic subspace of \(T\) is a closed infinite-dimensional subspace \(H\subset X\) such that any nonzero vector of \(H\) is hypercyclic for \(T\). The set of all these subspaces is denoted by \(\mathcal{H}(T)\) and called the hypercyclic spectrum of \(T\). The paper under review deals with the topic of hypercyclic subspaces, but in the setting of Fréchet spaces admitting a continuous norm. In his first proposition, the author gives a sufficient condition on a continuous linear operator \(T\) on such a Fréchet space in order to have nonempty hypercyclic spectrum. This result extends part of a previous statement of \textit{J.~Bès} and \textit{A.~Peris} [J.\ Funct.\ Anal.\ 167, No.~1, 94--112 (1999; Zbl 0941.47002)] to the setting of Fréchet spaces. Applications of these results to spaces of entire functions of several complex variables and of smooth functions are also given. The second part is devoted to the structure of \({\mathcal H}(T)\). In particular, given an operator \(T\) with nonempty hypercyclic spectrum and \(H\in{\mathcal H}(T)\), where \(H\) is not the whole space, it is studied when it is possible to obtain, apart from \(H\) and its infinite-dimensional closed subspaces, more elements in \({\mathcal H}(T)\). Finally, the main and most interesting result of this paper asserts that every infinite-dimensional separable Fréchet space \({\mathcal F}\) admitting a continuous norm supports a continuous linear operator \(T\) with nonempty hypercyclic spectrum. This statement has been obtained independently also by [\textit{L.~Bernal--González}, Proc.\ Am.\ Math.\ Soc.\ 134, No.~7, 1955--1961 (2006; Zbl 1094.47010)] and extends an earlier result by [\textit{F.~León--Saavedra} and \textit{A.~Montes--Rodríguez}, J.\ Funct.\ Anal.\ 148, No.~2, 524--545 (1997; Zbl 1028.47007)].