Here's a new paper-like background for free use on personal and commercial projects (this applies to all background patterns here).
Source V. Hartikainen
Paper model of a tetrahedron. Modelo de papel de um tetraedro.
Source laobc
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
Prismatic Hexagonalist Pattern No Background
Source GDJ
Zero CC tileable brick texture, photographed and made by me. CC0
Source Sojan Janso
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
Bright gray tones with a hint of some metal surface.
Source Hendrik Lammers
Fabric-ish patterns are close to my heart. French Stucco to the rescue.
Source Christopher Buecheler
Don’t look at this one too long if you’re high on something.
Source Luuk van Baars
Not so subtle. These tileable wood patterns are very useful.
Source Elemis
Prismatic Groovy Concentric Background No Black
Source GDJ
It was called Navy Blue, but I made it dark. You know, the way I like it.
Source Ethan Hamilton
From a drawing in 'Real Sailor-Songs', John Ashton, 1891.
Source Firkin
Remixed from a drawing that was uploaded to Pixabay by DavidZydd
Source Firkin
Super detailed 16×16 tile that forms a beautiful pattern of straws.
Source Pavel
Simple combination of stripy squares with their negatively coloured counterparts
Source Firkin
Remixed from a drawing in 'Sun Pictures of the Norfolk Broads', Ernest Suffling, 1892.
Source Firkin
Free tiled background with colorful stripes and white splatter.
Source V. Hartikainen
Prismatic Hypnotic Pattern No Background
Source GDJ
CC0 and a seamless pattern from a tile drawn in Paint.net .
Source SliverKnight