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
Derived from elements found in a floral ornament drawing on Pixabay.
Source Firkin
Abstract Arbitrary Geometric Background derived from an image on Pixabay.
Source GDJ
A free background tile with a pattern of pink bump dots. This background tile is sweet! Moreover, it's designed for use as website backgrounds.
Source V. Hartikainen
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
Prismatic Curved Diamond Pattern 7 No Background
Source GDJ
Detailed but still subtle and quite original. Lovely gray shades.
Source Kim Ruddock
To get the tile this is based on select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
Some dark 45 degree angles creating a nice pattern. Huge.
Source Dark Sharp Edges
The classic notebook paper with horizontal stripes.
Source Are Sundnes
The name is totally random, but hey, it sounds good.
Source Atle Mo
A seamless pattern formed from background pattern 102
Source Firkin
A dark one with geometric shapes and dotted lines.
Source Mohawk Studios
The name Paisley reminds me of an old British servant. That’s just me.
Source Swetha
A blue gray fabric-like texture for websites. An yet another fabric-like texture. It has subtle vertical and diagonal stripes to it.
Source V. Hartikainen
A comeback for you: the popular Escheresque, now in black.
Source Patten
Prismatic Geometric Pattern Background 2
Source GDJ
He influenced us all. “Don’t be sad because it’s over. Smile because it happened.”
Source Atle Mo
A seamless chequerboard pattern formed from a tile that can be had in Inkscape by selecting the rectangle and using shift+alt+i. Alternative colour scheme.
Source Firkin
Sort of like the Photoshop transparent background, but better!
Source Alex Parker