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Monday night noticed that steering wheel was moving back and forth. Probably less than 1/2" from CCW to CW. It starts at around about 15 mph and do not feel any movement of steering wheel at 50 to 60 mph.
Turns out tire pressure was way to low. After pumping up tires believed problem was gone. Then drove again and thought I just had wishful thinking but it was still not as bad.

Miniekie guy said tire balancing or alignment would not help. Have check out the suspension on the lift.

What should I look for when the car is on jack stands?

 
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do you mean in circular way?
if yes, causes can be several - suspension have a lot of components that can wear.
if it is in the direction of the car back and forth or to the sides, then it is truly worrisome.

 
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Yes, circular. The steering wheel turning slightly left and right.

 
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then, any rubber bushing could be suspect.
Any suspension connection point also, including the tie rods
and the steering box

 
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Inspect the steering system and see what's loose

 
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Thanks for the advice. Noticed steering wheel was moving again this past Tuesday.

Replaced the engine mount three years ago. Wonder if the transaxle mount needs replacing now.

 
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Engine/transmission mounts have nothing to do with play in the steering wheel.

 
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No movement at the wheels. Tie rods not loose.
Wonder about the CV joints and front disc brakes. Could a caliper be stuck?
Gas mileage has dropped each time I have refilled: 30, 28, 26 mpg.

 
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MPG drop could be O2 sensor on its way out (takes a lot for the light to come up)

Is the steering play still there?
Like said, any parts of the suspension/steering connection points could be to blame.

Brakes will not impact steering other than shake it if they vibrate too much.

 
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Guess I need to check the steering again.
Also noticed the car tends to drift left now.

The odd thing is this problem does not occur until 10 to 15 mph.
Then at 50 to 55 mph the problem seems to go away.

 
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take to an alignment shop, otherwise there might be early tire wear

 
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Appreciate all reply's in spite of my limited diagnosis aptitude.
Being slow of mind never checked the tires till this past week.
The right front tire has a bulge. Put the spare on and the problem is solved.