Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ0) is a singular MASA in M=L(Γ) which satisfies a weakly mixing condition. If, moreover, it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, Higman–Neumann–Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.