Is there any way to show the number of selected vertices in Edit Mode?
Here's a 2.8+ answer with some visual reference and additional info.
1. First, make sure Window > Show Status Bar is enabled
2. Next, to see the stats for selected geometry (Verts|Faces|Tris|Objects), Right Click on the Status Bar itself and enable Scene Statistics
These settings can also be accessed through Preferences > Interface > Editors > Status Bar (and here a contextual Video Memory option is listed as well).