Add a particle system to the object and set it up to make particles being emitted only from vertices.
Set up particle system
Add 2 objects, one - emitter, another to be emitted. Select emitter (sphere in your case) and add a particle system for it.
Leave it as Emitter, set amount of particles to be equal to vertices' number and toggle Emit from to Verts. Uncheck Random (or particles won't be put in place of every vertex rather than random ones):
Adjust settings of the particles which will represent vertices
You can adjust size of particles, set it to random, change location of all the particles, their rotation phase using Velocity and Rotation rollout options, use textures or vertex groups to adjust particles location even more.. and many other things:
Note that you may want to turn off particles' gravity if this object is animated so not to let them fall down.
Read more on particle systems: