There is no "best" each have there good & bad points & it depends on the shoot.
Paper - cheapest per square metre, widest range of colours, should be treated as a "consumable", give a flat clean colour. If used on location you need a hard smooth floor to stopp creasing & ladies heels going through or a sheet of hardboard underneath.
Vinyl - generally the moste expensive, limited range of colours, have a sheen to the surface, relatively hard wearing compared to paper, popular with event photographers where you have a high throughput of trafffic over the background, if they get dirty a bit of kitchen cleaner normally gets the marks off.
Muslins/cloth have a texture to them, cost wise in the middle, last a while, ( still use a few Lastolite cloths that are 20+ years old), if thin can be backlit & used in front of other backgrounds.
Then there are a whole range of theatrical cloths available from companies like J D McDougall. (their Black moulton is great if you want a black black )
Plus there is also the storage & portability aspect, a 9ft vinyl is fairly heavy & need a good solid support system, papr rolls lighter however if left permanently on support systems that don't have a central core or store badly can bow. "Fixed" widths ( papr rolls can be timmed using a wood saw eg if you want an 8ft roll, cut a foot off a 9ft one)
Cloths are easily stored, can be fixed in a vriety of ways & useful wher space is limited.
Most studios have a mix of all three.
If I could only own one background it would be a mid tone grey paper/wall & then assuming I had the space light it with gels see
here
