This paper discusses the property of local finiteness for t-norm monoids and extensions of them. The focus is (i) on t-norm bimonoids, which are of interest in the context of weighted automata, (ii) on those extensions of t-norm monoids which are reducts of t-algebras, i.e. of the basic semantic entities for the (fuzzy) logics of left-continuous and of continuous t-norms, and (iii) on extensions of t-norm monoids with their residuation based standard negations. The paper introduces also a kind of relativized local finiteness, and offers a short discussion of the finite model property for t-norm based residuated logics.