Oh, I can see this is a fun thread with great discussion!
Bill, on your AYSE set you lost very little in cracking that set out, mainly because you have the box and the fact that they were featured in your excellent book!
As far as methods of crackology I use the Vise method, with safety glasses AND Nitrile exam gloves (powder-free). Use a large clean vise that is securely mounted with a well lubricated spindle. Applying gradual pressure to the sides of the slab will usually result in the halves separating enough to extract the coin. Sometimes as Bill mentioned, a screwdriver is needed to help remove portions of the slab. I always use the gloves just in case I need to grab the coin if it decides to leave the slab unexpectantly.
Why do i remove coins from slabs you ask--For example, I am putting a presentation US Type set album together. I have elected to spend lets say $100 to $200 for each coin in order to reach the desired grades and or variety needed. You noted the presentation comment above. I am assembling a premium set that includes a 1853 arrows and rays half dollar, a 1931s Lincoln cent and so on. I am trying to purchase the majority of them raw but I won't let a good deal go by because it's in a slab. By shopping around at the major shows you should be able to buy somewhere in the middle of the wholesale and retail price even for the slabbed ones.
Why do I slab coins---1. Sometimes I slab a coin if it has a high enough value and will be placed in my collection for long term storage. I feel a slab gives it an edge if the storage conditions aren't perfect from a conservation point of view. I place my better coins w/slabs in archival sleeves and boxes prior to storage at the bank. 2. I also slab some coins / medals that I am de-acquisitioning from my collection that need a good marketing strategy. Usually these are higher value coins subject to counterfeiting or values that are grade sensitive and are being sold on the internet for example. I don't want my customer debating after the sale the grade or condition and the possibility of swapping out the coin and sending back a substitute. Well, they could debate it anyway but it does carry a little weight in my favor.
Never sell your coins you say? Well it may not be you, but rest assured that your coins will be sold someday. It is best to keep a good inventory and instruction on who to contact if you cannot handle the affair yourself. I have instructions to contact one of my coin buddies to assist with the liquidation and not to arbitrarily sell off to just anyone (including my buddy if you know what I mean). Why do I care at that point? Well, my heirs would be rewarded for having put up with a lot of coin antics such as coin magazines and books laying around, having to share the library with tons of numismatic literature, coin shows, storage issues and the like. You can use this to your advantage if subtle hints are dropped now and then as to the value of your collection and what they might receive if collection is sold wisely.