The need for slimes is clearly a desperate one for many, post 1.7. While I never had a particularly hard time finding them, some seem to have never seen them even now.
There are a lot of myths around about slimes, most notably that they spawn near water or lava. This is not true, although they are often found near lava, since both appear near bedrock. A lot of people have attempted to build slime farms at bedrock, to find nothing spawns!
The fact is, slimes only spawn in certain chunks, AS WELL as only being in layers 1-16. Some people know this, but many more do not.
So this means any old bedrock level room won't cut it, it has to be in the slime-eligible chunks. Finding them legit is a process of trial and error . . . the best bet is to make many large tunnels near bedrock and look for slimes spawning in them. If you do not mind using an outside program, however . . .
Trunkz has been so kind as to create a web-app AND a mod that will allow you to detect slime chunks, which you can find here, in his thread. Make sure to give him some rep for such an amazing contribution!. To make the most use of it, you want to find multiple chunks within close proximity. My personal favorite is a row of 3, but a 2x2 square, or even larger, could work. Keep in mind that the odds may be against finding large clusters together, so don't set your sights too high.
Here are some basic blueprints for my farm, and two screen shots.
The general rule is that you want to clear EVERYTHING from bedrock to layer 16 out of these rooms, to maximize spawning rates. Then, you want to install floors, which should increase the spawn rate. The nice thing is, large and huge slimes can spawn in small spaces, but instantly suffocate, splitting into smaller slimes. You can make each floor 2 spaces high, or 3 spaces high. I like fighting large slimes for fun, so my farm is 3 spaces high. The optimal floor placement is depicted in the blueprint.
Furthermore, you want to light each floor fully; this will decrease the spawning of OTHER hostile mobs, not only making each room safer to clear, but will increase the number of slimes that can spawn. Like other mob farms, its very important to not wait IN or next to the chamber. Create a waiting room about 25 blocks away; this will allow the slimes to spawn in the eligible chunks.
I have NOT designed an automated or grinder version of this farm. So far, I've noticed that the slimes LOVE to cluster in the corners. Once left to their own devices, all of the slimes in the chambers will huddle up in one of the corners of the room. Knowing this, some sort of drowning system could be quite easy, I am sure. Its very important to remember, however, that a slime will ONLY split if you do the exact amount of damage to kill it. If you deal more health than it has, it will instantly die, yielding no smaller slimes. Since small slimes are the only ones which drop items, you do not want to do this. Killing slimes with your fist or blocks is ideal.
I know this farm works. 2 days ago, I had about 500 slimeballs; now I have about 3500. Even a few hours with this farm should be more than enough for anyone.
I have yet to see slimes, but that's not too surprising, since I only have one relatively small room at bedrock. Once I dig out some more space I will give this a shot.
Congratulations on having probably the most useful thread on the front page at the moment!
Its really not too hard to find them, but you have to spend a lot of time at bedrock level caves. According to Trunkz' thread, only 1 in 10 chunks can spawn slimes. If you figure that in a network of caves, only maybe 5 in 10 chunks have ANY exposed area below layer 16, and then most of those 5 chunks is covered, add in the fact that the odds of a slime spawning when you happen to pass through is pretty small . . . well, it becomes pretty clear why they aren't so easy to find. I like to make a lot of excavation at lava level though, so I tend to find them a lot.