We perform a microscopic analysis of how the constraints imposed by conservation laws affect q=0 Pomeranchuk instabilities in a Fermi liquid. The conventional view is that these instabilities are determined by the static interaction between low-energy quasiparticles near the Fermi surface, in the limit of vanishing momentum transfer q. The condition for a Pomeranchuk instability is set by Flc(s)=-1, where Flc(s) (a Landau parameter) is a properly normalized partial component of the antisymmetrized static interaction F(k,k+q;p,p-q) in a charge (c) or spin (s) subchannel with angular momentum l. However, it is known that conservation laws for total spin and charge prevent Pomeranchuk instabilities for l=1 spin- and charge-current order parameters. Our study aims to understand whether this holds only for these special forms of l=1 order parameters or is a more generic result. To this end we perform a diagrammatic analysis of spin and charge susceptibilities for charge and spin density order parameters, as well as perturbative calculations to second order in the Hubbard U. We argue that for l=1 spin-current and charge-current order parameters, certain vertex functions, which are determined by high-energy fermions, vanish at Fl=1c(s)=-1, preventing a Pomeranchuk instability from taking place. For an order parameter with a generic l=1 form factor, the vertex function is not expressed in terms of Fl=1c(s), and a Pomeranchuk instability may occur when F1c(s)=-1. We argue that for other values of l, a Pomeranchuk instability may occur at Flc(s)=-1 for an order parameter with any form factor.