Summary: Let \(\Lambda\) be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of \(\Lambda\) with fixed dimension \(d\) and fixed squarefree top \(T\). Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of \(\Lambda\). In the case of existence of a moduli space – unexpectedly frequent in light of the stringency of fine classification – this space is always projective and, in fact, arises as a closed subvariety \(\mathfrak{Grass}^T_d\) of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety \(\mathfrak{Grass}^T_d\) is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of ‘finite local representation type at a given simple \(T\)’, the radical layering \((J^lM/J^{l+1}M)_{l\geq 0}\) is shown to be a classifying invariant for the modules with top \(T\). This relies on the following general fact obtained as a byproduct: proper degenerations of a local module \(M\) never have the same radical layering as \(M\).