Tip: Do not try to repair porcelain with finger nail polish or other porcelain paints if it is chipped, for neither of these will perform the function of filling the chipped area.
Solution: Wash the basin with TSP (trisodium phosphate) and rinse well with clear water. Wipe the chipped area with lacquer thinners. Follow safety instructions on packaging.
Mix equal quantities of epoxy putty (as per package instructions) in a quantity adequate to fill chipped areas. Apply into chipped cavity and smooth out with a finger tip. Dip your finger in soapy water first, this will give a smooth finish on the epoxy putty and it won't stick to your finger.
Allow to cure for the required time (epoxy puttys have faster curing times).
Using 400 grit wet/dry sandpaper, dampen area and rub gently until smooth leaving no ridge around repaired area. Take note of the manufacturers name on the basin (eg. American Standard or Crane) and purchase proper porcelain paint such as Porcefix or Porcepatch to match. Apply with included brush, allow to dry, rub with 400 wet/dry sandpaper as above and apply second and final coat.
Note: In the lower mainland of B.C. Profile Marble & Bath can cut out the sink (if it's a one piece unit with countertop) and insert a new porcelain sink in it's place.
Some plumbing companies are also able to do this, you might want to check for a reputable company in your area.