I just got this Elantra 2012. I notice that when I'm at a red light waiting to move, my foot is on the brake pedal, the car is shaking. It's not a hard shake, but just at my feet and in my seat I can feel it. What might this can be.
My first thought would be the motor mounts. They isolate the vibration of engine from the frame.
That just recently happened to my 2014 Chevy Sonic, turned out that it was a faulty O2 sensor that was making the computer add to much gas into the engine. I'd suggest getting it checked soon, I waited too long and had to rebuild my engine.