Symptoms for a bad strut, also known as a shock or damper, are oil on the strut, noises from one corner of the car and possible problems with the car handling on bumpy roads. The struts keep the tires on the road, even on uneven ground so that it has a better road grip.
Pushing down on each corner of the car should not cause any noises. If any of the corners make a noise, check for oil leaks on the strut. If after pushing three times the corner of the car does not return to its original position, this can also indicate an issue with the strut. If there is oil on the struts or the shocks, the oil that they require in order to work properly is leaking out and if this happens, they are bad and need to be replaced. When replacing one in the front or back it is good practice to replace the other front or back strut.
Even if there is no oil on the strut, there could be issues with the upper mount or bearing plate. If the bearing plate or upper mount is damaged, there may be a noise when turning the car wheels to the left or the right.