Is there any word which refers to the spilling, splattering, sprinkling or splashing of ink? (Or something less “violent”, like the pouring or dripping of ink?)
This sort of thing...
...is a blot. Or several blots of ink, in this case. Also inkblot.
Image reference: http://chainink.blogspot.co.uk/2010_09_01_archive.html
How about spatter? (Note, this is not the same as splatter.)
spat·ter (sptr) v. spat·tered, spat·ter·ing, spat·ters v.tr. 1. To scatter (a liquid) in drops or small splashes.
There are three common reasons why "traditional" ink ends up where you don't want it.
You accidentally knock over the ink-pot - normally described as spilling the ink.
Excess ink falls off the pen-nib en route from ink-pot to writing surface - you dripped the ink, resulting in a blot (noun), which you may blot (verb) with absorbent paper to stop it spreading.
An overloaded nib pools or blobs ink while you're actually writing.
But in practice today few people use quills or fountain pens where such mishaps can occur. Usually nowadays it's a cartridge pen or ballpoint, in which case the ink just leaks.
The first word that came to my mind was splotch.
One Ngram confirms that the word splotch is indeed coupled with the word ink in many contexts – more so than dirt.
Until today, however, I didn't realize that the word splotch also appears to be chiefly American.
Google images, upon getting a query for ink splotch, will suggest splatter, stain, splat, spot, and blotch as alternatives to be partnered with ink.