By virtue of the fact that we have a copy of the book “The NOT So Itsy Bitsy Spider,” I’d say it’s canonically itsy bitsy.
It appears that Amazon.com is neutral; it offers exactly 48 product hits for each version.
That would make for a rather boring series of comic books.
That’s not exactly an independently collected result.
Count me in with those who have always heard it as *itsy bitsy. *
Well, clearly there’s some natural selection at work here: Itsy-bitsy spiders seem to be achieving a dominant position in the waterspouts of the world, but pockets of eensy-weensy spiders are surviving. In the remote and isolated land of Ennzedd, home of the kiwi and the tuatara, there exists yet another living fossil: the incy-wincy spider. 
Another itsy bitsy here. When my kids were young we used to sing a second verse. With a very VERY sad voice, you sing “The poor little spider, it had NO place to go,” then pause and at a much faster tempo sing “So it climbed all over <child’s name> from her head down to her toe” while tickling the child. The contrast of the very sad voice followed by the tickle and laughter was a lot of fun.
Incy Wincy (possible spelling Eeensy Weensy). Itsy Bitsy is an American thing, I think.
Incy-wincy. It’s the only way to go.
Buh?
I get "Results 1 - 10 of about 39,600 for “incy wincy spider”. (0.14 seconds) "
Another itsy-bitsy here.
IIRC, column Dave Barry once had a reader poll exactly like this, and itsy-bitsy won in a landslide. I can’t seem to Google his exact quote, but the archetypal response he claimed to receive was something like “It’s the eensy-skeensy spider! It was always the eensy-skeensy spider when I was growing up in [name of dirtball state], and I am 137 years old.”
Lately my mind has been turning this into:
The * Mitsubishi* Spider
as if the Japanese introduced a new stylish, possibly water-climbing sports car.
Of course, that only really works if you’ve always heard it as “the itsy-bitsy spider”. Which is, I believe, the title of a short-lived animated series, too.
Bart Simpson sang “itsy bitsy.” That’s good enough for me.
I grew up with the understanding that it’s itsy-bitsy. However the only spiders I’ve noticed recently are the fighty-bitey variety.
Smartass. 
Re the OP, I say “itchy bitchy.” 
Achy breaky, perhaps?
We always said “eentsy-beentsy”. I wonder if anyone else remembers it that way.
I have no idea which one I actually learned first, growing up. and I was familiar with both versions. And I wouldn’t have known which one came to mind first except that the first few times my wife and I were launching into a duet I’d sing “eensy weensy” and she’d sing “itsy bitsy.”
I wonder if it’s regional, or what?
- Always itsy bitsy for me.
- That Joey Deluxe version of the song linked on the first page is awesome!
- I had no idea there was more than one verse to that song.