Can select polygon in attribute table but not in map canvas

Topology can be cleaned manually which is time consuming. Install the Geometry Checker plugin. Then via the menu: Vector >Geometry Tools >Check Geometries Select your layer and check the boxes No silver polygons under Geometry conditions and Check for overlaps and Check for gaps under Topology checks. Select how you want to fix the errors and click 'fix'.

Alternatively, you can use GRASS GIS v.clean if you have QGIS with GRASS GIS modules. Although I find that to be less intuitive.

This post also has good answers: Solving topological errors like invalid geometries in QGIS?

Indeed most of the time geometry errors make the features graphically unselectable.
You can edit the layer and resolve the errors by manually removing self-intersection or duplicate vertices.
Also, you can try some algorithms in the processing menu that may help but are not perfect :

• repair geometry ;
• delete duplicate vertices

At last, few tools may help you to check all geometries and identify errors like algorithm check validity(a name approaching) in vector menu or processing again.