Hello and thank you for the info.  Very much appreciated.
I have a 1965 Austin Cooper S and a 1967 Austin Cooper S.  Both have the remote shift design.  And both have the same vibration rattle or what ever which at first is strange but now thinking about it maybe not.

I do have access to a lift.  This weekend (maybe) I will do the dance on all the components mentioned.  I have had both autos on the lift before.  I do recall there isn't much room between the exhaust and the remote shifter aluminum housing.  I will look close at this area. 

Thanks much
Tim

To add to what 1963SV2 says, all Minis were built with an upper stabilizer (the 'dogbone') running from the engine above the clutch back to the firewall. As he said, its rubber bushings may be worn or its bolts may be loose or broken. Sometimes the brackets on the firewall also fail, though this is hard to spot until you disconnect the dogbone. If this is loose, the engine can rock, applying torque to the front subframe that causes the floor of the foot-well to oiican, eventually resulting in metal fatigue and cracking. This causes a creak, clunk or snap each time you go on or off the power.

'63 also mentioned differences in shift mechanisms. Rod change and magic wand shifters do not provide any bracing for the motor, bu a remote change type has a long cast aluminum housing bolted to the transmission and running all the way back to the gear shift lever, where it has a bracket securing it to the tunnel. It forms a very good engine stabilizer, unless that rear mount fails.
Once you have a look around at the underside of your Mini, you will see how little clearance there is between the exhaust pipe and the body, the shifter and the universal joints (or pot joints) on each side of the differential housing. It may be less than 1 cm in places. The exhaust system must be carefully positioned and well supported so that you can't move it by hand anywhere.

Hi Tim, I guess it depends on what kind of Mini you have... more especially, what kind of shift mechanism.

If you have big HP with a rod change box ... and use it competitively ... then perhaps you would see some benefit.  Otherwise........

With the handbrake on, reach in and grab the rocker cover and give it a good shove back and forward.. Does it move ??...at all?? You should not be able to move it with muscle power (assuming your middle name is not Arny  The car should move before the engine does....  If it does, carefully check the bushes at each end of the (standard) clutch end engine stabiliser. If these do allow more than the most minimal movement I would replace them first and then check whether you feel you need more stability.

As for finding your noise... if you can get the car on a hoist you give yourself the best chance of finding the problem.  Look for polished/shiny marks on exhaust or body; especially at the front end of the tunnel or in the vicinity of the inner drive joints.  Get under the car and give the exhaust a good shoving up/down/side to side ... does it move??) Even a tiny bit - there's not much room.  

If the car is dry, take a wheel off and give the shock absorber a similar heavy duty push and shove (back and forward) at each end... (Is there any movement in the bushes?)

Grab the drive shafts and see if you can move them up and down - or front to back ..... you will see some movement but not much.  In and out doesn't really matter.

While you're underneath you should be able to have a good look at the engine mounts... Do they look squashy/distorted??

Cheers, Ian

Hello.
I seem to have a vibration/rattle/thumping at the front RH side, at the floor, near the center tunnel.  I was thinking possibly the exhaust rubbing or my motor mounts are worn/broke/loose or something and was thinking of replacing the motor mounts to hopefully eliminate the noise.  I have no idea as to how to find the items that are vibrating, as I have looked.  Then searching on this site I found the stabilizers. 

I find on this site three engine stabilizers; one for the upper engine and two for the lower engine. 
Classic Austin Mini Engine Stabilizer Kit Upper Left Hand Drive
Classic Austin Mini Engine Stabilizer Steady Bar Kit Lower Right Hand Fly Side
Engine Stabilizer Steady Bar Kit Lower Left Hand (radiator Side)
Why are these necessary? If you read the blurb for each stabilizer, they are absolutely necessary for any mini and will prevent breakage of all sorts of engine components.  Really??  Am I to understand the original motor mount design was or is totally inadequate?!  I find this hard to believe.  Are these stabilizers more for the hard riders?  I would think the motor mounts would need to be "new" prior to installing the stabilizers, yes/no?

Would appreciate any input or comments or experience.

Thanks much
Tim