There are 3 zip codes in Columbia....21044, 21045, 21046. There is no "best" one, but in theory, if price reflects quality, the ranking would go from most expensive to cheapest as 21044, 21046, 21045. 21044 is more expensive because the western and southern sides of the zip. 21044 has River Hill, Columbia's most expensive area. It is also the newest...hence higher prices, newer schools, better shopping. 21044 also has the Hobbit's Glen golf course area. I also am fond of 21046. It is mostly single family homes and is in the best location to D.C.
Overall, all are very good, and I wouldn't shop just by zip code alone. I would recommend narrowing down your search first by price and style of home. Then see what different areas offer you. Also, don't forget about other areas of Howard County like Ellicott City (21043, 21042), Clarksville (21029) and Elkridge (21075). Many of these areas might offer you superior options.
Let me know if I can help you further.