  Classical spin liquids (CSL) lack long-range magnetic order and are
characterized by an extensive ground state degeneracy. We propose a
classification scheme of CSLs based on the structure of the flat bands of their
Hamiltonians. Depending on absence or presence of the gap from the flat band,
the CSL are classified as algebraic or fragile topological, respectively. Each
category is further classified: the algebraic case by the nature of the
emergent Gauss's law at the gap-closing point(s), and the fragile topological
case by the homotopy of the eigenvector winding around the Brillouin zone.
Previously identified instances of CSLs fit snugly into our scheme, which finds
a landscape where algebraic CSLs are located at transitions between \fragile
topological ones. It also allows us to present a new, simple family of models
illustrating that landscape, which hosts both fragile topological and algebraic
CSLs, as well as transitions between them.
