Alternative colour scheme. Not a pattern for fabrics, but one produced from a jpg of a stack of fabric items that was posted on Pixabay. The tile that this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
One more sharp little tile for you. Subtle circles this time.
Source Blunia
Seamless pattern formed from a tile that can be extracted by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin
Number five from the same submitter, makes my job easy.
Source Dima Shiper
Sounds like something from World of Warcraft. Has to be good.
Source Tony Kinard
The rectangular tile this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
A seamless pattern drawn originally in Paint.net by distorting a slice of background pattern 116 and copying the resulting triangle numerous times.
Source Firkin
One more updated pattern. Not really carbon fiber, but it’s the most popular pattern, so I’ll give you an extra choice.
Source Atle Mo
Remixed from a drawing in 'An Index to Deering's Nottinghamia Vetus et Nova', Rupert Chicken, 1899. The unit tile can be had by selecting the rectangle in Inkscape and using shift-alt-i
Source Firkin
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
A light gray fabric pattern with faded vertical stripes.
Source V. Hartikainen
Same as the black version, but now in shades of gray. Very subtle and fine grained.
Source Atle Mo
Remixed from a drawing in 'Incidents on a Journey through Nubia to Darfoor', F. Ensor, 1891.
Source Firkin