We shall discuss three problems in noncommutative harmonic analysis, which are related to endpoint inequalities for singular integrals on matrix or group algebras. In first place, we find a very much expected proof of the weak type $L_1$ inequality for matrix-valued CZOs which notably avoids pseudolocalization. It uses a new CZ decomposition for martingale filtrations in von Neumann algebras and a very simple but unconventional argument. In second place, we establish the weak $L_1$ endpoint for matrix-valued CZOs over nondoubling measures of polynomial growth, in the line of Tolsa and Nazarov/Treil/Volberg. These results solve two open problems formulated in 2009. An even more interesting problem is the lack of $L_1$ endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups, formulated by Junge in 2010. Given a locally compact group G equipped with a conditionally negative length $\psi: \mathrm{G} \to \mathbb{R}_+$, we prove that Herz-Schur multipliers with symbol $m \circ \psi$ satisfying a Mikhlin condition in terms of the $\psi$-cocycle dimension are of weak type $(1,1)$. Our result extends to Fourier multipliers for amenable groups and impose sharp regularity conditions on the symbol.