When I fire up the old 360 from '88 it makes a klunking noise. Just one thump at the moment the engine fires. Does anyone know this? I thought it might be a broken engine mounting rubber.

hi check that the exhaust down pipe is not hitting the cross member
if it is, like you say it will be the drivers side engine mounting
there are 2 types early are just rubber later are type of hydraulic filled with oil....hope this helps

i'd go with exhaust too - a broken rubber (snigger) will often lead to that sorta one off clunk

or a sticky starter pinion gear, sometimes when the starters are on the way out they stick when they disengage giving a slight clunking sound. If you've got the energy, worth getting the starter out and giving the gear a good greasing.

I don't think it is the starter motor. The klunk is to loud for that and the entire car wiggles when it makes the klunk. Can it hurt my engine if I don't sort the problem with the engine mounts out?