Simple gray checkered lines, in light tones.
Source Radosław Rzepecki
Simple wide squares with a small indent. Fits all.
Source Petr Šulc.
More in the paper realm, this time with fibers.
Source Jorge Fuentes
Heavy depth and shadows here, but might work well on some mobile apps.
Source Damian Rivas
A seamless gray background texture suitable for use on websites. To me, it has the look of stone. Feel free to modify it to meet your needs (by making it a bit lighter or darker, for example).
Source V. Hartikainen
Derived from an image that was uploaded to Pixabay by starchim01
Source Firkin
From a drawing in 'Prose and Verse ', William Linton, 1836.
Source Firkin
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
Black version of a pattern that came out of playing with the 'light rays' plug-in for Paint.net
Source Firkin
From a tile that can be had by selecting the rectangle in Inkscape and using shift+alt+i.
Source Firkin
Basket Fibers, Basket Texture, Braid Background style CC0 texture.
Source 1A-Photoshop
Same as the black version, but now in shades of gray. Very subtle and fine grained.
Source Atle Mo
A free seamless background pattern for use on websites.
Source V. Hartikainen
The tile this is based on can be had by selecting the rectangle in Inkscape and using shift+alt+i
Source Firkin
Carbon fiber is never out of fashion, so here is one more style for you.
Source Alfred Lee