I heard from a friend of mine that his son was asked to extend his mission (Southern Utah, I believe) by a few months, a couple of transfers is how he described it. Is this a fairly common occurence or is this a more recent thing due to a decrease in the number of missionaries?

When I was in Spain, there was a problem with sister missionaries getting visas, so a number of sisters were asked to extend their mission by one month, so there would be enough sisters. They didn't want to close areas and they needed sisters with experience to train the occasional sister who did get a visa. But that was a staffing shortage due to an international visa problem. I'm not sure why they would extend in Southern Utah or for more than a month.

Several Sister Mish I know tried to extend their mission, they were all denied.
A couple of the guys got their mission extended, they didn't ask for it and didn't want to but were guilted into stay longer.
One of the guys was registered for collage in Sept, they had him stay until October. His dad had a fit and I can't blame him.

I served in Cal. in 1993-94. One sister that was wuppose to go home same day I did, was asked to extend her mish for a couple of months. Some sisters had not shown up. They couldn't ask me because my visa said I had to leave the USA at a certain date which was precisely the one of my return thank god... I had been ready for my return 9 months previous ;-)

...of course, they couldn't wait to get rid of me so that's understandable. But it was rare in San Diego for a mission to be extended.

Alberta-Saskatchewan, 1971-73

It was part of mission lore that the companion of a companion of a companion of a companion somewhere back in the misty dawn of the human race had been granted a 6-month extension. Oooo, what a righteous elder he must have been to want to stay and work longer! So, once in a while, a kiss-ass or someone feeling guilty for wasting the previous two years would ask for an extension. Word would spread that he'd asked, which inspired copy cats. I don't think any were ever granted.