I agree with LtkernelPanic as ravines may happen to cut through a mineshaft or you might find one like i did mining down. Abandoned mineshafts are easy to get lost in and contain cave spiders which are very dangerous. Bring lots of food and equipment once you found one. Look out for chest that might contain gold, diamonds, or coca beans and mineshafts may intersect with strongholds or dungeons.
Honestly, the easiest way would be using an X ray texture pack and wandering around, keeping an eye out for the shape of mineshafts. Other than that it's luck. The world I'm currently playing in only has one Mineshaft near spawn, and it's under an ocean, I found it while boating around and noticed a strangely bright spot on the ocean floor.
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Do you have a lovely bunch of coconuts? Do you have them all standing in a row? Are there big ones, small ones, some as big as your head?
I agree it's mostly luck. I normally just stumble across them in caves. When you are in a cave or ravine area, look for any wooden planks or cobwebs. Those things in caves normally means abandoned mine shaft. Good luck looking