Was experimenting for learning sake on the vertex bevel in different scenarios like the 4 sided cone and stumbled across this.
If you create a cone w 4 sides using shift A -> mesh -> cone & set vertices to 4 in the cones properties panel , you can use vertex bevel on the top vertex of it (in this example) to flatten it. However, if you create a 4 sided cone using a cube by scaling the top 4 vertices to 0 to create the cone it doesnt work on that top vertex but does on the bottom 4. I also tried insetting the top face of a cube to 1 and scaling up on z axis to create a spike like effect to the top face of the cube and then tried flattening it using vertex bevel. Whether you use f6 or ctl shift b does not seem to work. Help please! Seems like you should be able to create spike like shapes on more than just the program created cones and then flatten them with vertex bevel. Wireframe (Z) doesnt help in this case for detecting geometry issues.