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
ZeroCC tileabel stone granite texture, edited from pixabay. CC0
Source Sojan Janso
This one resembles a black concrete wall when is tiled. It should look great, at least with dark website themes.
Source V. Hartikainen
From a drawing in 'Art Embroidery', M.S. Lockwood and E. Glaister, 1878.
Source Firkin
An abstract texture of black metal pipes (seamless).
Source V. Hartikainen
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
Prismatic Polka Dots Mark II No Background
Source GDJ
This pattern comes in orange, and it looks as if it is "made of glass".
Source V. Hartikainen
Dark blue concrete wall with some small dust spots.
Source Atle Mo
A seamless texture traced from an image on opengameart.org shared by Scouser.
Source Firkin
Like the name says, light and gray, with some small dots and circles.
Source Brenda Lay
Sometimes you just need the simplest thing.
Source Fabricio
From a drawing in 'Art Embroidery', M.S. Lockwood and E. Glaister, 1878.
Source Firkin
This background image is great for using in web design or graphic design projects. And don't forget to visit the homepage. I frequently update this resource with fresh tileable backgrounds.
Source V. Hartikainen
This tiled background comes in red and consists of tiles that look like gemstones. It is more for blogs or social profiles, I think.
Source V. Hartikainen
Formed from decorative divider 184 in paint.net. Vectorised with Vector Magic.
Source Firkin
A dark one with geometric shapes and dotted lines.
Source Mohawk Studios
Nasty or not, it’s a nice pattern that tiles. Like they all do.
Source Badhon Ebrahim
Bumps, highlight and shadows – all good things.
Source Badhon Ebrahim
From a drawing in 'At home', J. Sowerby, J. Crane and T. Frederick, 1881.
Source Firkin
Drawn in Paint.net and vectorised in Vector Magic.
Source Firkin
Seamless pattern made from a tile that can be obtained in Inkscape by selecting the rectangle and using shift+alt+i.
Source Firkin
From a drawing in 'Les Chroniqueurs de l'Histoire de France depuis les origines jusqu'au XVIe siècle', Henriette Witt, 1884.
Source Firkin