Stuck nut on bike axle?

The first approach to take is try to immobilize the axle by getting the other side dropout extra super duper pinched between the locknut and axle nut. If that side's axle nut can be freed, remove it and grease the threads. Snake in a cone wrench and open wrench on that side if applicable and get the cone and locknut very tight against one another. Then with a wrench on the cone if possible or the locknut if not, reef down on the axle nut. You may get more preload by adding a washer if it doesn't have one, and you'll tear up the dropout less. Then try again on the stuck nut. If you can put a wrench also on that side's locknut while doing so, it will help to immobilize the axle. (If you have access and think it will work, you could also just try putting a wrench on that locknut without doing the other steps. That will be enough some amount of the time.)

This only takes a minute and usually it just works.

Depending on what locknuts you do and don't have access to, another approach can be do the above to the extent possible, then try to break free the offending side's cone and locknuts from each other, then do wrenches on that side's axle nut and locknut.

Getting penetrating oil on the stuck nut will help with any of the above.