I am doing a modeling exercise and cannot select vertices alone.
The edges are always selected, too, but not fully, just as if they were more selected at the vertices. It’s as if the vertices bled into the edges.
This way I cannot do any bevel.
The whole model is compromised like this.
Where did I go wrong so no vertices alone can be selected?
Screenshot attached, thanks.
This is just how selected vertices display- don’t worry, you don’t actually have any edges selected 
That looks like you probably have some faces with flipped normals in there for some reason. You can check for that by setting Face Orientation in the Overlays:
Select the faces that show red on the outside and go to Mesh → Normals → Flip. Then your bevel should work as expected.
If you ever have a lot of red faces, it’s faster to select all and then do Mesh → Normals → Recalculate Outside.
This is my standard “cause #3 for weird stuff happening”. (Cause #1: scale not applied, #2: extraneous geometry present, usually from aborted extrusions.)
Actually, #2 could also look a lot like this. If you have extra vertices on top of only a few of the top ring, but you might have noticed that something was wrong already during selection. Still, if the normals are fine, select all, and do Mesh → Clean Up → Merge by Distance. That should get rid of the extraneous vertices.
