I would imagine that one or more shock absorber has failed/is failing. They tend to get get weaker with age.
Is the car sitting level? If not then check the springs to see if one has broken.
Can't see it being the top mounts to be honest, they do fail frequently but normally just cause noise and maybe a slight bit of movement, not enough to make you think your suspension has gone soft.
Other thing is tyres, some makes have very soft sidewalls but if you haven't recently changed them then it rules that out.
As for trying to pinpoint where the problem is, does it feel softer when cornering one way more than another? When braking does one side of the front dip down more than the other? Another good test for the rear suspension is to drive at a few miles per hour and slowly apply the handbrake firmly but not enough to lock the rear wheels, does one side of rear of the car dip down more then the other?
Good luck
