Schwinn Le Tour headset
 

I have a 1987 Schwinn Le Tour and the headset is a little loose feeling. I would like to take it apart to service it but I'm having trouble removing the nut. I have sprayed penetrating oil on it, but it wont budge. There are no signs of rust there. Could someone tell me if it could possibly be left handed threads. I don't want to break it. Please help.

It's not left handed threads.
Has it been in place since 1987?
What kind of wrench/tool are you using?

It's probably been in place since '87. I'm using a large open end wrench. I'm just afraid to put too much pressure.

Make sure you are only turning the top-most nut. The top race also has flat faces and the two are separated by a keyed washer. If you are putting the wrench over both nuts, then you are trying to turn that keyed washer which won't want to turn. Just a possibility.

Also, many headset top (lock) nuts have 8 flats, not 6, so a typical open end wrench won't get as much of a purchase as it normally would. As noted above be sure you are only trying to turn the top nut. The flats on the upper race are usually very thin and require a special "headset" wrench to fit them.

Using 2 headset wrenches (probably a 32mm, but could be 30/31/33/34mm as well) may make it easier- 2 leverage points to apply pressure to. To remove, you want to turn the locknut counterclockwise, like any other nut/bolt.
you won;'t strip threads with even severe amounts of pressure, but if the wrench slips off the locknut you could end up rounding out the flats. Avoid using a pair of channel locks, vise grips, etc.

Find a headset wrench to fit the lower adjustable portion of the headset and try turning it slightly clockwise to make sure it's not helping to bind the top nut. A high quality adjustable wrench - at least 12," preferably 16" will grip the top nut just fine. (Personally I have never been without a 16" adjustable wrench and a plastic dead-blow hammer. Headset wrenches tend to slip when applying extreme pressure. If that does not work carefully split the nut with a cutting wheel. You should eventually be able to pry the nut off in two pieces in a worst case scenario.

tap the flats of the top nut with a hammer. it will loosen any corrosion build-up.