I’ve just began using butter. My goodness. 
Duck fat
Darn, I don’t have that in the poll.
If I have bacon grease, I use that to fry potatoes in. I don’t often fry potatoes, and I don’t often have bacon grease, but when I do, the combo is awesome. Better than butter.
Bacon fat if I have it, but usually vegetable oil.
Canola, but I add spices to it first.
This. If cooking in veg oil, I usually add just a bit of butter near the end, just for flavor.
Butter and some red pepper oil.
Bacon fat or olive oil, depending on how lazy I’m feeling.
Vegetable Oil (I have used Crisco, as well). The rest of them scorch too easily.
That’s what our favorite restaurant uses. I’ve tried but the butter burns, so I switched to butter-flavored Crisco. It’s the only thing I use it for.
What kind of potatoes do ya’ll use? We use Simply Potatoes, fresh, already shredded.
I use margarine, so I couldn’t/didn’t vote.
You can also make your own by peeling a couple of potatoes and boiling them until lightly cooked. Heat your pan (cast iron works best), add butter/margarine/oil/whatever, and shred your potatoes directly into the pan.
You have to be careful not to over boil the potatoes because, if you do, they will just fall apart when you try to shred them. What I usually do is, if I’m cooking some for mashed, I’ll leave a couple of nice-sized ones whole and take them out of the boiling water about half-way through the cooking time for the ones for mashed.
Overnight in the refrigerator and you can have hasbrowns with your eggs in the morning.
Sorry. 
I’m ashamed to admit I’ve been using Kroger’s Frozen Country Style Hashbrowns.
Just because I know how to cook hash browns from scratch doesn’t mean I always do it. 
I have a bag of Inland Valley (brand) Simply Shreds in my freezer.
Sometimes convenience trumps “from scratch” and usually I only make mine from real potatoes if I’m out of the frozen or if I’m fixing mashed potatoes anyway and then I’ll boil a couple of extras.
Russets, usually. Yukon Gold work well, also. You want something with a lower moisture content so you don’t end up with a pile o’ mush. Try mixing butter with olive oil (or a neutral oil, if you don’t want the added flavor). That’s what I do for popcorn, as well.
A combination of butter and olive oil. I sometimes use Kroger’s country shredded, too. Most of the time I dice fresh red potatoes small, and halfway through the cooking add diced onion.
Hashbrowns means different things to people.
There’s the MCDonalds version. I’d usually call that a potato cake.
Dennys makes a very, very good shredded hash brown.
At home we dice the potato’s into small pieces. Fry in a pan with some ham chunks, onion, salt & pepper. Really it’s more pan fried potato than hash brown.
Bacon fat. Or if not available, then vegetable oil.