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
Nice little grid. Would work great as a base on top of some other patterns.
Source Arno Gregorian
A seamless pattern from a tile made from a jpg on Pixabay. To get the tile select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
A dark pattern made out of 3×3 circles and a 1px shadow. This works well as a carbon texture or background.
Source Atle Mo
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
To get the tile this is based on, select the rectangle in Inkscape and use shift+alt+i.
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
Scanned some rice paper and tiled it up for you. Enjoy.
Source Atle Mo
A criss-cross pattern similar to one I saw mown into a sports field.
Source Firkin
A repeating background of beige (or is it more vanilla yellow) textured stripes. One more background with stripes.
Source V. Hartikainen
A blue background wallpaper for websites. It has a seamless texture with vertical stripes. It looks quite nice not only when using as a tiled background on websites, but also on computer desktops.
Source V. Hartikainen
Sometimes simple really is what you need, and this could fit you well.
Source Factorio.us Collective
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
A seamless pattern formed from a tile made from ornament 22. To get the tile, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
A background pattern inspired by designs seen in 'Burghley. The Life of William Cecil', William Charlton, 1857.
Source Firkin
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
This is the third pattern called Dark Denim, but hey, we all love them!
Source Brandon Jacoby
More leather, and this time it’s bigger! You know, in case you need that.
Source Elemis
Seamless , tileable CC-0 texture. Created by my own, feel free to use wherever you want!
Source Linolafett
The square tile this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
Made by distorting a simple pattern using the 'sin waves' plugin for Paint.net and vectorising in Vector Magic
Source Firkin
Prismatic Geometric Pattern Variation 2 With Background
Source GDJ