I am looking for an easy way to create in Illustrator vintage style corners like on the picture provided.
I have some technical ideas, but they are time consuming, there should be an easier way I believe.
I would simply grab the Pencil Tool and scribble a rough edged corner, then use the Pen Tool to close the hard edges.
I'd do this a few times with varying colors and create several different corners, perhaps 6 or 8 corner variations.
The more detail you add to each of your individual corners the better they will ultimately work.
Then it's merely a matter of dragging the corners to the Symbols Panel to make them symbols and placing instances where you want the corners. Fill in between corners by again grabbing the Pencil Tool and scribbling fill-in shapes.
Made with just 2 corners, would look a lot better with 4 different corners.
While effects and brushes absolutely work, they often come across as "digital". By that I mean, they appear too uniform and patterned much of the time which removes any sense of human interaction. By manually creating corners you can reuse which are all different, you'll retain a great deal of the "human" aspect to any illustration/image.
It's just my opinion, but this extra effort is what separates great illustrations from just standard images.
I do realize this may not be the "easiest" way. But it is the method I would use.
You can use a brush to do this. For example prepare a corner, it does not have to be square. Then:
Image 1: A grunge corner made with this approach. I'm using one of the default brushes but you should have many more grungy alternatives available in your library.
You can use any grungy brush, they are also easy to make with a camera/scanner and tracing.
I place another answer due to some popularity of the question. The answers of Joojaa and Brian are good, however there are some issues with them:
Using a grundy brush "as is" (without creating a new brush) gives multiple islands detached from main shape and decrease the vintage effect;
Using Spatter effect creates raster image which has its own raster based limitations.
My half-minute solution is based on using Wrinkle tool - I played with options and it gave me result that probably other user would want to achieve: there are less islands than using brush (I place a screenshot where I did not adjust anything), secondly the shape remains vector as I need.