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
Free tiled background with colorful stripes and white splatter.
Source V. Hartikainen
From a drawing in 'Sun Pictures of the Norfolk Broads', Ernest Suffling, 1892.
Source Firkin
Prismatic Abstract Geometric Background 2 No Black
Source GDJ
From a drawing in 'Codogno e il suo territorio nella cronaca e nella storia'', Gio and Giarella Cairo, 1897.
Source Firkin
Tiny little fibers making a soft and sweet look.
Source Badhon Ebrahim
Prismatic Curved Diamond Pattern 3 No Background
Source GDJ
From a drawing in 'Az Osztrák-Magyar Monarchia irásban és képben', 1885.
Source Firkin
Seamless Dark Grunge Texture. Here's a new grunge texture for use as a background.
Source V. Hartikainen
A seamless pattern formed from background pattern 102
Source Firkin
Made by distorting a simple pattern using the 'sin waves' plugin for Paint.net and vectorising in Vector Magic
Source Firkin
A seamless paper background colored in pale yellow.
Source V. Hartikainen
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
Seamless SVG vector and JPG backgrounds with faded diagonal stripes. The colors are editable.
Source V. Hartikainen
This was submitted in a beige color, hence the name. Now it’s a gray paper pattern.
Source Konstantin Ivanov
The tile can be had by using shift+alt+i on the selected rectangle in Inkscape
Source Firkin
This one needs to be used in small areas; you can see it repeat.
Source Luca
This one is rather fun and playful. The 2X could be used at 1X too!
Source Welsley
Carbon fiber is never out of fashion, so here is one more style for you.
Source Alfred Lee
From a tile that can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
Here's a new gray "fabric" pattern. Use it as backgrounds for websites or for other purposes.
Source V. Hartikainen