About 2 weeks ago exactly the same thing happened to me.
Sitting in front of my iMac 27" (late 2013) I heard a snap or crack and the screen tilted forward. My heart stopped for a moment and after several expletives and cold sweats I established all was ok except the stand. Thanks to this post and other similar ones I was relieved to find out it was an uncommon fault - but not unheard of.
The computer was just out of its 12 month warranty and I thought f%$K! - this is going to cost a lot. Various figures mentioned on blogs ranged from $24 - $600. Having had bad and frustrating experiences with Applecare in previous years I thought they would refuse to accept liability for a parts failure so soon after the 12 month warranty ran out. The computer which had sat on the same desk since new had failed and I imagined Apple not accepting liability for something that could only be a parts failure or defect. Fortunately they were actually very helpful on the phone and agreed in principal to fix it for free. By this stage I remembered that EU (European Union) consumer law effectively means your products should be covered for 2 years (for pre-existing faults) and Apple also made it clear on the phone that I had additional consumer protection under EU law even though their parts warranty had recently expired. They booked in a genius appointment and wrote up notes effectively saying to the store that it should be fixed for free.
Although I didn't have to pay for the repair they have to cost it. So I got to see a cost for the repair. I thought this figure would be huge given the work involved (un-glueing the screen and taking the whole thing apart). It was about £50 inc VAT - £22.80 parts and £28.80 labour. The Apple guy said they no longer have different rates for labour in the way they used to. It's now a standard flat-fee no matter how much work is involved.
If it weren't for additional EU consumer protection it's difficult to know if Apple would have accepted liability and done the repair for free.
Excellent and speedy work carried out by Apple Store, Cribbs Causeway, Bristol, UK. Computer back up and running and stand working fine - let's hope it doesn't happen again 12 months from now - from various other posts it does seem to happen around the year-old mark