Since there seems to be a bunch of ways to build Lulu, I am wondering which one is the best for domion(ap,ad, on hit, support, etc)?

In my opinion, support.

I've seen AP, AD, and on-hit. Do they work? Sure...

...but that's like going to Ruth's Chris Steakhouse and ordering a cheeseburger and fries.

but that sounds like a **** good cheeseburger. lol

OP, i think tanky cdr support is the way that most play her nowadays. (thats the way most supports are played to my knowledge, except for perhaps sona).

Sona is played tanky cdr also, the only difference is that sona actually gets a tri-force, the tanky cdr build consist on Ionian,Guise,Fh and then go full tank

I personally play her as OnHit similar to Kog Maw. She lacks Kogs range of course and her onhit just isn't quite as strong (87 max, vs 120 vs a 2k opponent). But she does have the following advantages:

a) The AP on Malady and even Nashers tooth are way more useful on her, than Kog.

b) Her onhit damage is permanently up, vs Kogs limited time duration.

c) Her spells are MUCH better, especially for an autoattacker's role. Her slow is incredibly strong, better than Kogs, she has the nasty Disable, and a self-shield/ranged nuke. her Ult is also ideal for CCing the typical enemy melee-er and boosting her survivability.

d) She benefits strongly from CDR due to her nice CC, making CDR items more viable (like nashers tooth as a 3rd item potentially).

This role requires you to play her like you would an AD carry. She ramps up much much faster than an IE/PD carry though, and brings her own CC.

Usual build of: Zerker boots, dagger to start, Malady, Wits End, Stinger, Phage, Ionic Blaster doodad (now buffed), ->Nashers tooth, -> Frozen Mallet.

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The other way to do it is to tank up like crazy. An Ionic Boots, Frozen Heart maxes her CDR and lets her spam CC like few others, plus making her very hard to kill. Stack on the Odin, an Aegis, and a giant belt or two and she greatly contributes as a support, except she doesn't evaporate when focused like most of them...

You could build on-hit, and it's fun, but you'll fall off lategame damage-wise and be far too squishy. I feel it's generally better to build tanky cdr and donate your on-hit buff to somebody else on your team.