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
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
The original enhanced with some gradients.
Source Firkin
Remixed from a drawing in 'Line and form', Walter Crane, 1914.
Source Firkin
Geometric triangles seem to be quite hot these days.
Source Pixeden
Design drawn in Paint.net, vectorised using Vector Magic and finished in Inkscape.
Source Firkin
Prismatic Geometric Tessellation Pattern 2 No Background
Source GDJ
Zero CC tileable brick texture, photographed and made by me. CC0
Source Sojan Janso
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
To get the tile this is based on, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
A new one called white wall, not by me this time.
Source Yuji Honzawa
Pixel by pixel, sharp and clean. Very light pattern with clear lines.
Source M.Ashok
Simple gray checkered lines, in light tones.
Source Radosław Rzepecki
An alternative colour scheme to the original seamless pattern.
Source Firkin
Remixed from a drawing in 'Paul's Sister', Frances Peard, 1889.
Source Firkin
Prismatic Basic Pattern 2 No Background
Source GDJ
More carbon fiber for your collections. This time in white or semi-dark gray.
Source Badhon Ebrahim
It almost looks a bit blurry, but then again, so are fishes.
Source Petr Šulc
From a drawing in 'A Life Interest', Mrs Alexander, 1888.
Source Firkin
This seamless background image should look nice on websites. It has a dark blue gray texture with vertical stripes, it tiles seamlessly and, like all of the background images here, it's free. So, if you like it, take it!
Source V. Hartikainen
This pack of filters can help you adding a blocky overlay to objects. May come handy at drawing blocks of stone.
Source Lazur URH
Hexagonal dark 3D pattern. What more can you ask for?
Source Norbert Levajsics
A seamless pattern with wide vertical stripes colored in pale yellow.
Source V. Hartikainen
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