They have a discrete spectrum, which you can see just by holding a diffraction grating close to your eye, and looking to the top or bottom, away from the light. You will see a sequence of distinct monochromatic images of the exposed portion of the bulb. A deep violet, a deep blue, an aquamarine, a green, then a fainter spread of yellow to red, with a distinct bright red image embedded in it.
I just checked this with my diffraction grating which I keep handy at all times :-)
We recently got LED streetlights, and their spectrum is broken, but has long continuous ranges. It's still qualitatively different from the household incandescent spectrum, which is continuous, and peaks in the infrared, which gives it a bright red tail. I can still see plenty of green, blue, and violet, though. ( Spectral violet looks "purple" to me, although technically purple is a mixture of spectral red and blue. )
Hmmm. Somewhere, I have a pocket spectroscope. I need to try that.