System Information
Operating system: Windows-10-10.0.19042-SP0 64 Bits
Graphics card: GeForce GT 740/PCIe/SSE2 NVIDIA Corporation 4.5.0 NVIDIA 456.71

Blender Version
Broken: version: 3.1.0 Beta, branch: master, commit date: 2022-02-21 23:16, hash: rBec5bbebf3ee1
Worked: 3.0.1 Official

Short description of error
While in edit mode, mesh becomes unselectable when Toggle X-Ray (not wireframe viewport shading) is active if mesh selection mode is set to Faces when objects have modifiers where Show On Cage and Show On Edit Mode options are active (i.e. subsurf modifier). Note you can still press A to select all in this state and you can select while verts or edge selection modes are active but not Faces.

Exact steps for others to reproduce the error

1. Open Blender.
2. Add a subsurf modifier to the default cube.
3. Enable the "Show On Cage" setting of the subsurf modifier.
4. Enable Toggle X-Ray (not Wireframe viewport shading).
5. Change mesh selection type to Faces.
6. Try to select with any Blender selection method, no selection methods work while Face selection mode is active.