This is a fix I made on a "medium grade" rug, made with rows of stitching. Nicer plush rugs may not work for this.
It also helps if the rug is a little multicolored to help hide your repairs.
I'll go over a small and large snag, with photos. I already just about finished the large one when I decided to document it.
Flexible fabric glue (I opted for some that dries clear and flexible)
Straight pins (or something similar, for the large pulls that yield long pieces of string/yarn/fiber)
Scotch tape (as needed)
Something heavy like a text book
Step 1: Trim
Step 2: Glue
Use the tip of the glue pen or your finger to work the snag into the hole.
If possible, work the surrounding fibers into the hole to help cover it and look more full.
Step 3: Set
My glue stated 2-4 hours dry time, so I just left it over night.
Now for the large snag...
Step 4: Oh noes... a big one.
This snag was about a foot long and the yarn-like fibers came to be about a couple feet long (hard to tell, they're stretchy). It unravelled along a single row of stitching. It has a orange and green straight pin at either end in the first photo.
I placed the straight pins at either end and wound the yarn about the pins until it looked like it filled in the void nicely. I taped off the excess incase I needed more as you can see in the second photo.
I pressed the tip of the fabric glue pin between the wound section and the normal, carpet section and applied a line along both ends for the entire length. I also added a generous amount around either pin because I was paranoid of it coming undone when the pins came out.
I also had to add some glue here and there because the top layers weren't touching any glue. I basically created a semi-flexible patch in the rug using the pulled out yarn. While it doesn't feel exactly like the rug around it, it is flexible and won't stab bare feet or anything.
I put wax paper and a book on top as stated before in the same fashion and inspected my results the next day. Trim excess as needed, remove tape and pins.
Step 5: All done!
This photo contains the repaired long snag and a few small ones like in the beginning (Sigh... Lilli the Beagle...)
While not ideal it is much cheaper than a new 6x8 ft rug and since it is in my computer room/guest room I'm quite happy with it.
Hope this helps!
PS: Bonus Tip.
Bleach spots on the bathroom rug? Keep a multicolored pack of permanent markers on hand to color in those bleach spots.