Semiclassical analysis and noncommutative geometry are distinct fields within the wider area of quantum theory. Bridges between them have been emerging recently. This lays down on operator ideal techniques that are used in both fields. In this talk we shall present semiclassical Weyl’s laws for Schrödinger operators on noncommutative manifolds (i.e., spectral triples). This shows that well known semiclassical Weyl’s laws in the commutative setting ultimately holds in a purely noncommutative setting. This extends and simplifies previous work of McDonald-Sukochev-Zanin. In particular, this allows us to get semiclassical Weyl’s laws on noncommutative tori of any dimension $n\geq 2$, which were only accessible in dimension $n\geq 3$ via the MSZ approach. There are numerous other examples as well. The approach relies on spectral asymptotics for some weak Schatten class operators. As a further application of these asymptotics we obtain far reaching extensions of Connes’ integration formulas for noncommutative manifolds. (For Riemannian closed manifolds, Connes’ integration shows that Connes’ NC integral recaptures the Riemannian measure.)