You might want to keep in mind that I've been out of the pro game for a while, and that I tend to shoot people "in the wild" these days, so I'm not entirely up to date on the latest offerings. I was really hoping to see some other input here from people who have more recent experience in a professional studio environment, but here goes:
Canvas
Canvas is extremely durable, but it's not particularly portable or storable. As a painted fabric, it's well-suited for surface cleaning (wiping with a cloth or sponge); as a dyed fabric you need to allow for light colours to become somewhat dingier over time and for darks to fade -- dry cleaning, as opposed to laundering, helps, as does dark storage in a sealed bag. White isn't very white unless the canvas is painted (bleaching only gets you so far), so you need space to light the background separately if you want anything that's apparently white. (We'll take it as read that if you want "digitally white", you need to overexpose any background.)
Canvas can be fairly heavy; even the lighter grades of cotton duck sold as "canvas" are in the 10-12oz. region and painted finishes add considerably to that, so you need a sturdy support (not necessarily a certified and approved official background stand set, but a couple of thumbtacks aren't going to do it). It takes and shows wrinkles and creases, although they will relax into near invisibility over time if the canvas is dyed and unsized (and can be ironed out). Painted canvas really needs to be treated like roll paper; the binder in the paint (even if there is no visible paint film) makes creases and wrinkles very prominent. It's great stuff for swappable scenics or a durable seamless, but they really need to be kept either hung or rolled (and may take days of hanging to become usable if they were folded, even around a form, when you purchased them).
Synthetics may have changed the game a bit since I last looked; I can imagine that something other than dropcloth-grade cotton duck (which was the base fabric for all of the lighter canvasses I've worked with) might be somewhat better-behaved, but I haven't seen any evidence of them in my local pro photo emporia.
Muslin
I don't know what the kids are doing with muslin these days, but in my day it was the location alternative to mottled canvas backdrops. Not only is muslin much lighter and slightly cheaper, it takes wrinkles better. Yep, that was the point. Getting canvas to a location meant either bringing an unwieldy long roll of fabric with you, folding it neatly and having a regular pattern of creases (which look like a bad web page background image), or having a very prominent spider web of irregular creases. Canvas is a heavy fabric; it only scrunches up so much. Muslin, on the other hand, is very lightweight and glories in the ability to take an almost fractal irregular creased texture if you just sort of scrunch it up and stuff it into a bag. When that texture is combined with the soft mottling that's usually applied, it just sort of disappears. If you want a plain background, then you need to treat muslin with almost the same care as canvas.
Muslins do tend to be made from a higher grade of thread than canvasses (the source looms and threads are used for things like bed linens rather than industrial applications and grounds for painting on when they're not doing their twenty minutes a year of photographic applications). That means that the results of bleaching and dying tend to be better -- you can get a good white and intense, bright colours without painting, as well as darks with no sheen. But wrinkling and creases are always going to be a problem if you need to store or move them.
Jersey (Knits)
Jerseys tend to get around the wrinkling and creaing problem very neatly, and because of that they make great solid-colour backgrounds. Being fabrics, of course they are going to wrinkle and crease, but those wrinkles and creases can be stretched out easily. But like the man said, there ain't no such thing as a free lunch.
Jersey's don't just hang there; they pretty much need to be stretched. If you put a plain jersey background panel on a standard background stand, the top may be the advertised width, but it's going to taper down to this little six-inch wide cylinder at the bottom. You need to clamp the fabric to the uprights in oder to maintain width, and the clamps need to be frequent enough to avoid creating a visible diamond pattern due to variations in the tension. And it's pretty much a background-only material; it doesn't do well as a seamless, even if you try to stretch and weight the perimeter of the piece on the floor/table. (The stretchiness means that there's almost always a bubble popping up somewhere.) It makes a fantastic pop-up, though, and it works great on a frame -- wrinkles just vanish.
And there's the pilling problem. If you never need to clean the fabric, it can stay looking new for a very long time. If you do need to wash it, you need to remember that you're dealing with something that's really just an oversized tee shirt. Washing it by hand in cold water in the bathtub by squeezing (no agitation) and hanging to dry works well, but anything rougher than that will pretty much ruin it. If it's something that going to experience a lot of rough-and-tumble, it's the wrong fabric.
Velour/Velvet
These make great solid colours (assuming we're not talking about the "crushed" variety) and there is no substitute for black velvet as an absolute black apart from immense distance -- light goes in and it doesn't come back out. The nap hides wrinkles and creases in the base fabric very well. For lighter/brighter colours, you may need to brush the surface of the nap in order to avoid apparent colour changes (one of those large blackboard erasers that's essentially a foam rubber block with a stick inside of it and a chamois glued over one face is very quick and effective for this). Velvet/velour is great for solid whites, solid absolute blacks, and for chromakey-type colours, and a white or grey panel combined with gelled lighting can make for very smooth gradients. Velvets can be surprisingly light (although some, made as coating fabrics -- that is, fabric suitable in weight to be used for coats -- and those based on a jersey knit rather than a weave, can be surprisingly heavy).
Velvets and velours, though, are lint magnets, and the lighter colours can pick up handling grime quickly. They are absolutely not suitable for seamless applications where anything heavy (like a person or even a large tabletop subject) is involved -- raising the nap again after it has been thoroughly crushed is a bear of a job.
There's a huge range of washability -- the name really only tells you what the surface of the fabric looks like; it doesn't tell you much about the underlying construction. Some will go bald if you look at them funny, some will felt, others will laugh at your "hot" water and puny washing machines. Some will dry quickly and easily, others will absorb Lake Superior in a single gulp and be ready to use again next February if you're lucky. It's probably safe to assume that any company that depends on the goodwill of professional photographers (Lastolite, Photek, Photoflex, Westcott, and so forth) will sell you something suitable for the purpose; but as with anything else in life, I'd be wary of anything priced at the "too good to be true" level -- it may work well out of the box, but it's probably disposable.
Really, Truly White
If you are working an a small space (no real room for separately lighting the background evenly from the front) and need something that's digitally white rather than merely apparently white (that is, something that doesn't merely look white to the viewer, and can have shadows, but that will actually be all Fs when you look at the values), then it may be worth looking at backlighting. Commercially, that means using an enormous softbox or something like Lastolite's Hilite background system (which is essentially a shallow sidelit softbox). A DIY version wouldn't be too difficult to create -- the only piece that's critical is the diffusion panel, and that's really only "critical" in the sense that it needs to be seamless (one piece). The rest is just reflective fabric (that can be the cheapest silver lamé you can find at the fabric store, and can be full of seams everywhere) and a frame to hang it on (PVC, anyone?).
