The cheapest and easiest way to replace a bathroom sink in a mobile home is to buy a sink made for mobile homes from a mobile home supply company. They are inexpensive ($10-15) and come in various colors. Make sure you ask for the drain hardware too; the counter guys at the parts places frequently forget to suggest it.
If the faucet is leaking or causing problems this is a good time to replace it too. You may be able to find replacement gaskets at a parts store, but it is usually much simpler to just buy the faucet new.
The sink is held in place by clips that are screwed in place around its edge. You will get new clips with the new sink. It is such an easy job, it is not worth trying to polish an old one or fighting to remove burn marks or scratches from an old one.