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
Prismatic Abstract Geometric Background 2
Source GDJ
From a drawing in 'Cowdray: the history of a great English House', Julia Roundell, 1884.
Source Firkin
A seamless pattern formed from a sports car on clker.com. To get the tile, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
This one resembles a black concrete wall when is tiled. It should look great, at least with dark website themes.
Source V. Hartikainen
f you want png files of this u can download them here : viscious-speed.deviantart.com/gallery/27635117
Source Viscious-Speed
The tile this is based on can be retrieved by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
Seamless Prismatic Pythagorean Line Art Pattern No Background. A seamless pattern that includes the original tile (go to Objects / Pattern / Pattern To Objects in Inkscape's menu to extract it).
Source GDJ
To get the tile this is based on, select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
Prismatic Hypnotic Pattern No Background
Source GDJ
People seem to enjoy dark patterns, so here is one with some circles.
Source Atle Mo
Prismatic Polyskelion Pattern No Background
Source GDJ
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
The unit cell for this seamless pattern can be had in Inkscape by selecting the rectangle and using shift+alt+i.
Source Firkin
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
A slightly more textured pattern, medium gray. A bit like a potato sack?
Source Bilal Ketab