Don’t use scouring pads. Use either Bon Ami or Barkeepers Friend. They won’t scratch and erode the porcelain finish.

If it is cast iron the brown could be rust stains try peroxide to remove.

Your sink is wearing out and needs to be replaced. I see it all the time.

Thank you for your comments. For my next sink, should I be wiping using a cloth with BarKeepers Friend?

I'm also guessing there's no way for me to apply a new finish to the sink?

There is no refinishing of porcelain sinks that lasts that I've ever seen. You'll get plenty of offers from the spray-on guys, but several months later the coating begins to fail at the drain flange which they didn't bother to remove. The money you gave them would have made a nice down payment on a replacement sink.

If you get another porcelain sink get the grids for the bottom to prevent damage from pots etc.
I put mine in the dishwasher (top rack) almost every time I run it.

should I be wiping using a cloth with BarKeepers Friend?

You sprinkle the BarKeepers Friend powder lightly around the sink. You do not let the powder stand, but immediately, with a wet sponge, wipe the powder all around with the wet sponge to clean the sink. Then rinse thoroughly with water using your faucet's spray function.

Your wet sponge must be just a sponge, not an abrasive sponge, not a scrubby sponge, not a scouring sponge. Just a sponge.

Ajax has bleach and bleach oxidizes cast iron. The brown is rust leaching onto the porcelain. Soak a towel in peroxide and place it on the brown areas for a while and see if they lighten. Beach is not appropriate for a cast iron tub or sink.

Thank you very much for the advice. Much appriciated <3

No need to buy a new sink, have it re-glazed or refinished which is done in your house.

The spots are from hard water and supposedly we're supposed to wipe the sink dry each time we use it. Riiight.

Barkeeper's Friend (powder) is abrasive, not sure why people say it's not. After I bought my sink I used Kohler's special cleaner for porcelain & I wasn't that impressed.

"No need to buy a new sink, have it re-glazed or refinished which is done in your house."

Terrible advice. I've never seen the reglazing work more than a month or two. The reglazing guys don't even remove the disposal flange which is where the coating begins to peel. The $400.00 they're going to charge would have made a nice down payment on a sink replacement.

The only time a reglaze on a sink makes sense is if your house is for sale and you no longer live there.