  Classical spin liquids have been a very active subject of research in the
last few years. A very rich variety of cases have been shown to exist,
including short-range and algebraic spin liquids displaying dipolar
correlations at zero temperature. In this article, we investigate different
families of classical spins liquids by combining analytical techniques and
Monte Carlo simulations. Our study relies on the Luttinger-Tisza approximation
(LTA), which is associated with the constraint vector function in momentum
space, whose general properties allow for a classification of different spin
liquids. We show that the general properties of the LTA provide a framework for
identifying and accurately characterizing the different types of spin liquids
in different geometries. We apply our approach to three different families of
spin liquids defined on the checkerboard and kagome lattices, which exhibit a
remarkable range of situations, including various cases of algebraic and
short-range spin liquids. Additionally, we analyze the effective Gauss law
emerging from different kinds of spin liquids and identify states that exhibit
additional degeneracy lines. The presence of spin-liquid phases and pinch-point
singularities are confirmed by Monte Carlo simulations validating our approach.
Our study opens up avenues of research in the study of spin liquids, exploring
algebraic spin liquids with higher-rank gauge fields and as critical points
dividing different types of classical spin liquids.
