I took the liberty of using Dmitry’s pattern and made a version without perforation.
Source Atle Mo
Seamless pattern formed from a tile that can be had by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin
A free background pattern with abstract green tiles.
Source V. Hartikainen
Heavy depth and shadows here, but might work well on some mobile apps.
Source Damian Rivas
To get the tile this is based on, select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
The tile this is based on can be retrieved by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
The image depicts a seamless pattern which includes hexagonally-aligned gourds with BG in light-brown.
Source Yamachem
Sharp but soft triangles in light shades of gray.
Source Pixeden
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
Remixed from a drawing in 'Incidents on a Journey through Nubia to Darfoor', F. Ensor, 1891.
Source Firkin
Simple gray checkered lines, in light tones.
Source Radosław Rzepecki
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
Zero CC tileable yellow craft paper; scanned and made by me. CC0
Source Sojan Janso
Seamless pattern formed from a tile that can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
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
Seamless Prismatic Geometric Pattern With Background
Source GDJ
Sounds French. Some 3D square diagonals, that’s all you need to know.
Source Graphiste
Retro Circles Background 5 No Black
Source GDJ
Recreated from a pattern found in 'Az Osztrák-Magyar Monarchia irásban és képben', 1882. To get the tile this is based on select the rectangle in Inkscape and use shift+alt+i.
Source Firkin
Zerro CC tillable texture of stones photographed and made by me. CC0
Source Sojan Janso
Greyscale version of a pattern that came out of playing with the 'light rays' plug-in for Paint.net
Source Firkin
Looks like a technical drawing board: small squares forming a nice grid.
Source We Are Pixel8
To get the tile this is based on select the rectangle in Inkscape and use shift-alt-i
Source Firkin