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
Not so subtle. These tileable wood patterns are very useful.
Source Elemis
Remixed from a drawing in 'Analecta Eboracensia', Thomas Widdrington, 1897.
Source Firkin
Colour version of a pattern that came out of playing with the 'light rays' plug-in for Paint.net
Source Firkin
ZeroCC tileable wood boards texture, photographed and made by me. CC0
Source Sojan Janso
This is so subtle I hope you can see it! Tweak at will.
Source Alexandre Naud
Prismatic Chevrons Pattern 5 With Background
Source GDJ
Prismatic Abstract Geometric Background 5 No Black
Source GDJ
Prismatic Geometric Tessellation Pattern 2 No Background
Source GDJ
A seamless pattern that includes the original tile (go to Objects / Pattern / Pattern To Objects in Inkscape's menu to extract it).
Source GDJ
Seamless pattern formed from a tile that can be had by selecting the rectangle in Inkscape and using shift-alt-i.
Source Firkin
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