The first pattern on here using opacity. Try it on a site with a colored background, or even using mixed colors.
Source Nathan Spady
Here's a quite bright pink background pattern for use on websites. It doesn't look like a real fur, but it definitely resembles one.
Source V. Hartikainen
There are quite a few grid patterns, but this one is a super tiny grid with some dust for good measure.
Source Dominik Kiss
More bright luxury. This is a bit larger than fancy deboss, and with a bit more noise.
Source Viszt Péter
A light background pattern with diagonal stripes. Here's a simple light striped background for you.
Source V. Hartikainen
Drawn in Paint.net and vectorised in Vector Magic.
Source Firkin
From a design in 'Storia del Palazzo Vecchio in Firenze', Aurelio Gotti, 1889.
Source Firkin
A tile-able background for websites with paper-like texture and a grid pattern layered on top of it.
Source V. Hartikainen
Thin lines, noise and texture creates this crisp dark denim pattern.
Source Marco Slooten
The green fibers pattern will work very well in grayscale as well.
Source Matteo Di Capua
From a drawing in 'Les Chroniqueurs de l'Histoire de France depuis les origines jusqu'au XVIe siècle', Henriette Witt, 1884.
Source Firkin
From a drawing in 'At home', J. Sowerby, J. Crane and T. Frederick, 1881.
Source Firkin
Inspired by this, I came up with this pattern. Madness!
Source Atle Mo
The act or state of corrugating or of being corrugated, a wrinkle; fold; furrow; ridge.
Source Anna Litvinuk
From a drawing in 'Les Chroniqueurs de l'Histoire de France depuis les origines jusqu'au XVIe siècle', Henriette Witt, 1884.
Source Firkin
Inspired by a design found in 'Konstantinápolyi emlékeim', Miklos Chriszto, 1893.
Source Firkin
Derived from a corner decoration itself found as a jpg on Pixabay.
Source Firkin
Remixed from an image on Pixabay uploaded by Prawny
Source Firkin
Colour version that is close to the original drawing uploaded to Pixabay by pencilparker.
Source Firkin
Alternative colour scheme. 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
This is a semi-dark pattern, sort of linen-y.
Source Sagive SEO
A version without colours blended together to give a different look.
Source Firkin
Prismatic Groovy Concentric Background No Black
Source GDJ
Vertical lines with a bumpy, yet crisp, feel to it.
Source Raasa