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 design seen on Pixabay. The basic tile can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
A free light orange brown wallpaper with vertical stripes designed for use as a tiled background on websites. An yet another background pattern with vertical stripes.
Source V. Hartikainen
A grid of squares with green colours. Since the colours are randomly distributed it is automatically seamless.
Source Firkin
To get the tile this is based on, select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
Remixed from a PNG that was uploaded to Pixabay by VictorianLady
Source Firkin
The green fibers pattern will work very well in grayscale as well.
Source Matteo Di Capua
Remixed from a drawing in 'Works. Popular edition', John Ruskin, 1886.
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
The original has been presented as black on transparent and stored in the pattern definitions. To retrieve the unit tile select the rectangle in Inkscape and use shift+alt+i
Source Firkin
Prismatic Snowflakes Pattern No Background
Source GDJ
A repeating background of thick textured paper. Actually, it turned out to look like something between a paper and fabric.
Source V. Hartikainen
A free seamless background with pink spots.
Source V. Hartikainen
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
A seamless pattern the unit cell for which can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
An abstract Background pattern of purple twisty patterns.
Source TikiGiki
This is so subtle you need to bring your magnifier!
Source Carlos Valdez
Remixed from a drawing in 'The March of Loyalty', Letitia MacClintock, 1884.
Source Firkin
More in the paper realm, this time with fibers.
Source Jorge Fuentes
Not strictly seamless in that opposite edges are not identical. But they do marry up to make an interesting pattern
Source Firkin
CC0 and a seamless pattern from a tile drawn in Paint.net .
Source SliverKnight
It’s like Shine Dotted’s sister, only rotated 45 degrees.
Source mediumidee