Okay, here's the deal. Savages don't go off by themselves from the factory. They are an inerently safe design that is among the sturdier actions one can find that have a superb system of venting gas away from the shooter's eyes should a case head fail.
However, the trouble can come in when people start playing with the trigger. The trigger can be adjusted well, especially with a Timney trigger. However, as with all rifles, when you do a trigger adjustment, you have to ensure that you haven't adjusted it too light with too short a pull. Without scope mounted (just because there's no need to jar the scope), you should always perform a bump test with any trigger adjustments. Adjust the trigger, put the rifle back together, and cock it on an empty chamber. Then butt the rifle against the floor, butt downwards. If the trigger is unsafely adjusted, it may fire on its own as the shock is enough to jar the sear and pop the trigger loose. Some folks don't do this and end up with a rifle that is not safe to carry.
I have not personally heard about this with Savages, and have never read any documentation what so ever about it happening on factory rifles. Heck, I've never heard of it happening save for a warning about adjusting the trigger. Savage went through financial troubles because they had too many products and were stretched very thin. The finish on their 110's was weak and the stocks were really cheap looking. Never the finest looking rifles, the 110's became down right Blue Light Special in appearance. They have always been accurate, but some of the 110's, especially the E models, just looked like crap. Indeed, they didn't really look any better than the 320's, which really looked cheap, and the 99's were just too expensive to produce and make much profit on. The markets responded with people buying better looking Remingtons and the like.
Remember, except for Weatherby, none of the major firearms companies we know of today are owned by the founders or are really even remotely related to them save by name only. All of them went through major financial troubles at one time or the other and were bought and sold with some frequency. Smith and Wesson was owned for a time by a British company known as much for plumbing supplies as firearms. Remington was once owned by Dupont, and we all know about Winchester almost going belly up and now being made by US Repeating Arms.
Savage is doing well because the 110's accuracy reputation has finally reached the level where it should have been. Also, Savage has streamlined its production to a level where they are not so overextended. Most of all, though, it was bought and (I think) privatized by folks who have a passion for succeeding. Most established companies go through trouble not because their products are horrible, but because management loses vision and the lost energy very clearly shows up in the product, the advertising, and the like.
I would not hestiate to own a Savage, regardless of year of production. Both of mine are pre Accutrigger and are fine rifles. Both have adjustable triggers, one, a short action 110 in .243 (before they were called 10's) sold under the Stevens name, has a factory adjustable trigger whereas my 111 (which I recently rebarrelled to .270) has a timney in it. If I needed another rifle right now, I would not hesitate to get an Accutrigger, which is supposed to be utterly safe against the butt test. I am satisified with mine and as I have all the calibers covered (7mm Mag, .30-06, .308, .270, and .243), I haven't felt the need to upgrade.