I was looking at my last Harvey Milk stamp and wondering how easy it would be to just print up your own batch of stamps. They do not seem to have any security features, compared to printed money. In its noble history stretching back to Ben Franklin, Has the USPS ever struggled with people selling fake stamps?
I cannot comment on the question of whether there has ever been an "outbreak" of counterfeit stamps, but as regards:
They do not seem to have any security features
This would be an incorrect assumption. There are lots of security features on stamps, mostly revolving around micro-printing techniques, invisible inks, laser etching, and other similar technologies.
There has periodically been some news of an artist using the postage stamp as, well, a medium for art. I am trying to recall the earliest one- I think he was active in the 1970's, did some very witty ones for imaginary countries: one set from a recently-liberated former colony with smiling children, one from a grim dictatorship with stern, proud soldiers. A more recent one is Michael Thompson, but there have been others Here's a book on the subject.
The bottom line is that there's too much work in making a fake stamp for it to be a problem, the market value of the art stamp being a hundred times the face value. It would be like someone making 25-cent coins out of solid gold. Not a big danger.