What's the most efficient Minecraft mining strategy?

Mining

Branch mining is the most efficient. There's a mathematical analysis on it in the official forums (the link is to an archive of the post).

Here's a brief description of branch mining:

• You dig down to the bedrock, because that's where you're going to find the most different kinds of stuff.
• One or two layers above the bedrock, so you don't have it interfering with your work, you dig a long tunnel, 2 high and 1 wide, which will serve as the trunk.
• Then, every 4 squares on either side of the trunk, you dig perpendicular tunnels extending out 20 blocks (or as far as you want).
• You leave 3 undug blocks between because veins are generally 2x2 horizontally, so you will see ore in the vein even though there is technically one row of unexplored blocks.
• When you finish that layer, you can start one a layer above, offsetting the branches by one block so that you are more likely to find veins you could possibly have missed the first time.

So essentially, a long hallway (the trunk) with perpendicular hallways (the branches) coming off either side.

██▒▒▒▒▒▒  ██ blocks you should mine
██▒▒▒▒▒▒  ▒▒ blocks you can see
▒▒▒▒██▒▒  __ blocks you can't see if you don't mine ▒▒ blocks (there's none!)
▒▒▒▒██▒▒  Repeat this pattern as many times as needed

From a head on view

Travel

As with any mine, travel is most easily accomplished using minecarts.

You can place a booster to get your cart traveling at max speed in a short distance so that you can make the trip up quickly, and you can even attach chests to mine carts to carry all your findings from the mine up the track very quickly.

Short of minecarts, make sure you're using stairs and ladders to their fullest potential. You don't want to be jumping up one block at a time all the way from your mine to your base.

There is a great wiki with some number crunching.

I use this in single player:

▒▒██▒▒__▒▒__   ██ blocks you should mine
▒▒██▒▒__▒▒__   ▒▒ blocks you can see
__▒▒__▒▒██▒▒   __ blocks you can't see if you don't mine ▒▒ blocks
__▒▒__▒▒██▒▒   Repeat this pattern as many times as needed

Normally, I have a 2x1 trunk, and branches like a standard branch mine, but instead of just going outwards from the trunk, I go up one level and out, skip two blocks, then down one level and out. I leave two blocks between branches. And I leave two blocks between floors, and line up the branches the same between floors.

There are some areas left unexplored by this pattern, but they are thin, and it is unlikely that a diamond vein with spawn entirely within that narrow region. (About 3% will). The larger number of blocks uncovered are far more likely to contain diamonds.

If your high density branch mine gets you 100% diamonds, the low density branch mine would give you 212.6% ores in the same time, minus 3% from the earlier probability worked out before.

Also, you ought to build it so that the floor of the lowest tunnel is on level 11, to avoid falling in lava. If you find that there are few cave systems near your mine, feel free to dig lower, but caves tend to cluster, and caves level 10 and below are filled with lava.

I dig my main tunnel on both sides of the area I want to cover, then dig back and forth in a zigzag patten, which means I'm always digging, never walking.

I'm not sure if this is true anymore, but it used to be that larger deposits were formed by the word generator in multiplayer, so you were less likely to miss diamond deposits when using a wider mining pattern. Thus, this pattern

▒▒██▒▒____▒▒____  ██ blocks you should mine
▒▒██▒▒____▒▒____  ▒▒ blocks you can see
__▒▒____▒▒██▒▒__  __ blocks you can't see if you don't mine ▒▒ blocks
__▒▒____▒▒██▒▒__     Repeat this pattern as many times as needed

is more efficient.

As for transit? I use a long two wide straight staircase that goes all the way to bedrock. Minecarts work well for getting me deep into the mine.

If you're not married to building a completely man-made mine, the answer is to find a deep tunnel.

Getting down to the bottom is correct, as that is where the highest concentration of useful minerals is located, but digging a tunnel doesn't expose you to nearly the same amount of blocks as using a generated cave system.

In my primary game, I dug a mine down to the bottom, clearing out a large amount of stone, which only resulted in some coal, and very few iron, where as at the bottom I discovered a cave network (filled with lava, but some water solved that), which resulted in a large amount of coal and iron, as well as diamond and gold.