Which stamps are not valid for postage? You may be surprised!
To prevent use of stamps by the Confederacy, the USPOD printed new stamp designs in 1861 and invalidated all earlier issues to prevent their use, especially by Southern states, for the payment of postage. This is one time when people holding stamps of the older designs were allowed to exchange their stamps at post offices for stamps of equal value in the new designs.
Some issues are very obvious for not being valid for postage: Official Mail stamps, revenue stamps (including Duck Stamps), saving stamps, parcel post due, special handling, and postage due issues. Some services (such as special handling) are no longer a valid class of mail. Since the service is not offered any more, the issues for that service are not valid for postage.
Although the United Nations is headquartered in New York, UN stamps are not valid on US mail.
Here is something that will surprise you. Special Delivery and Certified Mail stamps are also not valid for postage. However, you still see these issues used as postage and nothing is said. Why?
The later Special Delivery issues (#E20 to #E23) and the Certified Mail stamp (#FA1) are older, but still modern looking stamps. The denomination is clearly visible. Many people working in the post office today were very young (or not even born) when these stamps became invalid for postage. However, most postal clerks are unaware of this. Chances are, you can still use them for postage and no one is ever going to notice.
Also invalid for postage are precancelled postage stamps or many of the service inscribed stamps (Bulk Rate, Presorted, etc.). You can use these stamps under certain circumstances. Their use requires a permit though.
Use of some stamps for postage would be ridiculous though. Technically speaking, any of the 1869 Pictorials (#112-#122) are valid for postage. Anyone who uses one of those issues for face value postage needs to have their head examined by a professional.
Some older stamps can be used for postage though. Take the 2¢ Washington (Scott #634) and other common and inexpensive issues from the 1920s and 1930s. They are sometimes used for postage. These issues have minimal catalog value. Sound copies are very common. If you have a copy that has a large thin or missing corner, good luck trying to sell it for a nickel. No one wants damaged copies when sound copies are cheap and plentiful. In those cases, yes, using a #634 for postage is perfectly fine because any additional collectible value is near zero.