The axe I split with the most is an ugly old no-name double-bit that kicked around my office for nobody knows how long. It's pitted and worn and there are no marks remaining to tell who made it. I think it was originally a Western pattern but it's so chewed up that that may be wishful thinking. It did not want to be handled, really, but I did it anyway -- the eye was dented badly enough that it was pretty tough to get a handle to stay put. I did clean up both edges a bit so that they were at least parallel corner-to-corner. This axe sits on my front porch with a no-name pickaroon and a slab I ripped out of a blowdown fir for a splitting surface. Neither edge is sharp. It splits like a dream.
My favorite felling tool is a chainsaw. I'm no Luddite.
My favorite axe for cutting is a 50's-era Plumb double-bit cruiser that I found in our warehouse. It sits under the bench seat in my work rig. I have several axes that are prettier or less worn and which would easily command bigger eBay dollars but this thing is just nice. Every time I find a log across a road, the decision I have to make is "PPE up?" If yes, I gotta gas/oil the saw, get into chaps, find earplugs, all that. If it's small enough and under little enough tension that I can avoid that, safely, I'll trade the extra effort of an axe for the extra effort of getting geared up for one or two cuts. Similarly, if I expect more blowdowns and I'm already geared up, I won't take anything off. My truck is pretty dusty.
I am always ready/willing to buy whatever axe catches my fancy. If it's $10 or less and I recognize the name and it's not totally worn out, I'll probably buy it without question. If it costs more than $20, I'll want to think about how much work it will take to bring it to my meager standards, and what I will use it for. If it's more than $50 It better have a super-cool story or be in perfect condition. I passed on a very nice-looking Puget Sound pattern falling axe for $50 yesterday because it was pitted enough to obscure the maker's name, it had no handle, and the temper lines were no longer visible. I bought a similar axe just a few months earlier for $10 and have yet to hang it on a handle. I surely don't need another of the same thing which also comes with regret.
Long/short -- buy lots of inexpensive old axes. They'll mostly lack handles. Learn to hang them. Learn to use them. Find out what it is that you like. Buy more of that.