Way late to this discussion but, for general purposes, stock or OEM equipment on any car - especially the suspension - is ALWAYS crap unless you're buying a "performance car".
Next mod for my'13 Elantra Coupe is lowering springs and shocks.
I bought my car in winter. Just before a huge dump of snow, and I do a lot of commuting on a barren highway. Roughly 130kms roundtrip.
The first drive into town and the rear end swung out laterally on me, or it felt like it was going to, just like everyone is saying. I complained to the dealership about the random fishtailing and said I didn't feel safe, but they (even to this day) refuse to do anything for me.
I bought better tires, thinking the ones they sold me were crap "all-weather" tires... very little difference. At this point I hit the internet and found MANY other Elantra Coupe owners had the same problem.
TL;DR - I have a PierceMotorsports rear torsion bar on the way, and am holding out for a used set of coilovers to throw on... after reading this, I'm comforted to know that at suspension swap out will resolve the issue, or at least improve it significantly.
Just so you're aware, this issue still isn't resolved on the 2014 Elantra platforms... :/