
Square dinnerware is the craze when it comes to modern dinnerware. At first, I didn't like it, but it has grown on me. Now, I like square plates and platters, but I'm still hesitant on square cups and cereal bowls. It just doesn't seem right, but each to their own, right?
Evidently some of the contemporary dinnerware producers agree with me because I have seen some sets that have round bowls and coffee mugs with square plates. I see myself choosing those rather than going "totally square".

Some people are thrilled with odd shaped dishes and it does complement contemporary design. A lot of the major brands of casual dinnerware are producing square sets - Corelle, Corning and Pfaltzgraff for instance.
I haven't seen very many in fine dinnerware, but Mikasa does make a bone china platinum pattern that is quite elegant. The cup is round, but the bowl and plates are square.
The Martha Stewart Collection also has a square pattern called Avenue Square. It is porcelain. The plates and the bowl are square but the cup is round and it has a comfortable handle (something that coffee lovers [like me] are adamant about).

If you like different shapes of dinnerware, I'm sure you would find the Martha Stewart Collection Scalloped Whiteware to be attractive.
Some antique dinnerware have scalloped shapes and I've seen some odd shaped platters and serving bowls, but square plates are rare in antique sets. That is more of contemporary dinnerware style.