What vowel sound do you make when you say "bird" in addition to the R sound, out of curiosity? I just make the R sound.
Trouble is, I don't speak IPA and whatever word I compare it to someone will say that differently too... Ah, internet to the rescue. At
http://www.thefreedictionary.com/Birdclick on the Union Jack with speaker, that's close enough to what I say. [ETA: nb, it's not "in addition to" the R sound; I have a non-rhotic accent, that is, I don't say an R sound in "bird"]
There are probably a lot of words out there, e.g. "purpler", "rattling", etc., that contain syllables without any vowels / vowel sounds no matter how you define such things. I'm guessing intuitively, as an overly opinionated layman who's too lazy to do real linguistic research on the weekend, that there would be a lot of words with multiple vowel sounds in the same syllable too-- what about "quiet"?
DH says "well I don't know, depends how you want to define it!" - but says it's either a dipthong (one vowel, one syllable) or two syllables, each with one vowel. It depends a bit on accent again (most people will say "fire" is one syllable, and for some people, it's the same sound) but from the sound of it more on which definitions you're following today.
it certainly does look like an interesting field, in a nitpicky sort of way.

Yup!