Tsai-Wu and similar criteria don't have details of the failure. You can look at each component of the criterion for clues and it pays to recover the three basic (1, 2 and 12) stresses and strains in each layer. If the layers are UD carbon/polymer then it will usually be clear what is likely to be breaking first in the layer which is predicted to be failing, fiber or matrix. The components of Tsai-Wu have to be recovered manually (or in a spreadsheet), so it's often easier to output the basic stresses and strains.
If your fiber is not carbon then it will likely be less clear. It will also depend on how brittle the matrix is. Glass has a tensile strain to failure of between three and four percent, which is close to many polymers, although it is likely to exceed them a bit.
It can also be hard with woven material. With cloth/polymer essentially in-plane shear close to the allowable is an indication of matrix failure. BUT, with cloth if the fiber is carbon it can be harder to tell than if it's glass, as carbon's in-plane shear failure stress can be close to that of a matrix resin, although it is usually higher. HS and IM carbon can be assumed to have a shear strength of 17 ksi/120 MPa (this is old Amoco data), annoyingly close to many polymers. Glass's shear strength should be in excess of 200 ksi or 1300 MPa (glass fibers are more-or-less isotropic and will have a shear strength above 50% of the tensile strength, and UTS is probably between 2200 MPa and 2800 MPa).
There are a few ways to keep an eye on matrix and fiber separately but none work well (you can model a layer of resin separately from a layer of fiber but this sort of thing has a lot of drawbacks).
Keeping the three basic stresses and strains for each ply available is sensible, usually pretty easy and passably useful, especially for UD carbon/polymer. (It also hays to keep an eye on through-thickness shear, which is usually available, certainly with Nastran and PCOMPs/CSHELLs, which I assume you are using from your mention of a 'solution sequence 101'.)
