When I put the rear wheel into the frame dropouts of the bicycle and tighten the wheel nuts up on the outside of the frame dropouts, the axle may turn and cause the cone lock nuts to come loose since they bear directly against the inner frame dropout surfaces.The bearing cones also come loose and the rear wheel wobbles. There are flat washers between the outside wheel nuts* on the outside surfaces of the dropouts on either side. Should there be a washer between the inner dropout surfaces and the cone lock nuts as well?

Should the cone lock nuts have blue Loctite on the threads to keep them in place?

* the nuts that actually mount the wheel on the frame.

You do have a two wrenches to fit the cone and locknut, right?

If your Schwinn is like what I just went through with my wife's 7 speed Schwinn, the cone that slides inside the sprocket cluster needs to be tightened hard with its locknut before sliding the axle back into the hub. Then install the other cone and locknut after installing the axle, then set the cone to the bearings, then tighten that cone's locknut. Really tight on both lock nuts or the cone just spins in or out depending on what you're doing when you install the wheel into the rear drop outs.

Of course, the wheel has to spin as free as reasonable without perceptible wobble or you have to readjust the cone again.

Your lock nut isn't tightened against the cone tight enough to LOCK IT! Not much purpose in having a LOCK NUT if you don't operate it properly.
Chase the threads on the axle with a die nut. The outside nuts should spin on with your fingers.

Is Loctite on the cone lock nut threads still ill-advised? Yes, I know that you have to hold the cone with a cone wrench to tighten
the lock nut with another wrench and I do have all those correct tools.

I didn't know that lock nuts have to be torqued like the devil.

Should there be no flat washers between the dropouts and the lock nuts?

Normally your lock nut is serrated on the DO side to grip the DO.

The outside nuts are rather sticky. I'll have to check my tap and die set for the right size die for the chasing.

When tightening the outside nuts, should I tighten each nut alternately a little at a time when snugging the wheel to the frame? I think my bearings came loose when I tightened one side too much before tightening the other side of the wheel.

OK, I see my troubles. I need to turn my lock nuts around. The smooth sides of the lock nuts are showing not the serrated sides.

I had the wheel apert for lubing the bearings two months ago. I may have assumed the serrations were to make the lock nuts grip the cones.

I'll see if I can't clean my axle threads up with a die while I'm at it.

Thanks, people.

No Loctite. Just operator error

Actually, there were no serrations on the other sides of the lock nuts either. This is the replacement wheel I bought from Pacific Cycle two months ago with all its axle hardware of its own. I kept the custom hi-grade solid axle I bought in 2018 and the original lock nuts and axle hardware from the original Schwinn-branded wheel. Those original lock nuts do have serrations on one side. So while I was at it I put the custom better, stronger axle on to boot. Threads are much better quality and nuts screw on easily with fingers. The threads were bad on the axle coming with the PC replacement wheel and that axle was noticeably warped also. I replaced the original 24-spoke stock wheels on my Schwinn Discover with aftermarket 36-spoke PC wheels. I gave my 24-spoke stock wheels to Goodwill not long ago. For safe keeping, I took out the custom axle and original axle hardware from the original rear wheel, however, before giving it to charity.

You mean mechanic's error. In terms of vehicles, fixing it and driving/riding it are two different things.

How often have you goofed working on your own car? I have a Hayne's book for my car, however.

I wish there was a Haynes/Chilton's for bikes.

Loctite is a last resort for many bike situations. A crutch instead of solving the problem. Bill gives good advice.

One trick I learned decades ago is to "counter tighten" the axle nuts as you secure the wheel. That is to snug one side's nut down to only "hug" the drop out. Then tighten the other side a tad bit more. Return to the first side and snug down more. back to the second side and a bit more tightening. Continue until both sides are properly tight. If the wheel isn't centered well the fully loosening one side and using the other side's, now, vice like clamping to control the axle's rotation as you center the wheel with the freely moving loose side.

With a well locked together cone and lock nut the axle should remain stationary (WRT rotation/spinning) within the cone/lock nut. By watching the axle/locknut for the VERY BEGINNING of any rotation and STOPPING the tightening of that side's axle nut you don't let the axle/lock nut/cone more relative to each other (the cause of bearing adjustment shifting from intended). As soon as the first side's tightening produces axle rotation you need to stop and go to the other side.

Two axle nut wrenches on both nuts and both tightened at the same time could also work but then there's more to pay attention to and more to not track and react to. Andy

At one time there was. Andy

Thanks Andy and Bill. There was no axle rotation at all this time with the lock nut serrations now biting the DO's of the frame. Snug on one side, then snug on the other, flipping sides until both sides are tight. On my bike, the wheel is perfectly centered when the axles are bottomed out in the dropouts on both sides: the axle ends are all the way home in the frame PROVIDED the axle is not bent which my good hi-grade steel solid axle is not.