I have a 1994 Corolla, which when idle vibrates, I've change the engine mounts last week and was fine then but after bumping the front left hand side today started vibrating again. Engine mounts are intact but when in Drive mode it does vibrate quite a bit. When it goes into Neutral gear it doesn't though.

SOURCE: toyota corolla 1.6 1997 vibration after clutch change

Try checking the engine/transmission mounts, one of them may not have been reinstalled properly, or worn out, causing a vibration. If the car is able to move with the new clutch, and it seems to work, I wouldn't say the clutch was put in wrong. Also check the wires around the engine/transmission, you may have knocked something loose as well.
Hope this helps

SOURCE: vibration through car

If you don't have to be moving to get the vibration then it has to be either something is not balanced properly (torque converter may be bad, front pulley, engine flywheel cracked) or you have a bad mount, causing the vibration from making the engine or trans contact the car body in a place you can't see.
Hard to pinpoint without feeling it myself. I can say for sure it's not the cam belt, unless there's debris stuck to the inside of the cam gear causing the vibration.

SOURCE: Excessive vibration of the steering column

Sound more like out of balance tires, or bent wheel. Unless this vibration is even in park. Then I would say something is up against a part of the steering that shouldn't be.. Or your exhaust system is bent or touching part of the undercarriage that it shouldn't be. Was the car ever wrecked?

SOURCE: My 2005 Corolla XRS has rattling when deaccelerating

I wouldn't say it is common. There are quite a few heat shields around the stock exhaust "header" and exhaust system. There are also FOUR motor mounts that you can check to see if all bolts are tight. They are all 14mm heads on the motor mounts, 12mm on the exhaust shields. These seem to be the main things that may get loose or make a rattling sound. The exhaust gasket is also designed to flex when the engine flexes - I would check into the exhaust. You have to remove the TRD strut bar to gain better access from above.

SOURCE: Toyota corolla engine vibration after changing the clutch plate

Toyota corolla engine vibration after changing the clutch plate