Question: I have painted my object with vertex-color on vertex-paint. How can I make the color visible in object-mode or edit-mode.

Meaning, how do I implement the vertex-color onto my faces so that I can see them on object-mode or edit-mode?

Answer: 
In 3D View, hit `N` to bring up the right panel, then enale Textured Solid.

![Textured Solid](https://i.stack.imgur.com/Ox8GH.png)

Now in Object Mode and Edit Mode, you will see the vertex colors.

![enter image description here](https://i.stack.imgur.com/weVOJ.png)

EDIT: This appears to only work with Blender Render.
The vertex color is show in **3D View** depending of your **Viewport Shading**

One menu next to where you choose *Edit Mode*, *Object Mode*, *Vertex Paint*, etc., the icons look like spheres showing different textures. You can switch between them. To see your vertex paint, **Texture** is needed.

If you need vertex paint on your renders, it's another story  
(*Vertex Color Paint* in *Properties Editor > Materials > Options*).