Good question, probably killed more with a handgun than rifle over the past 20 years or so. For a rifle several friends use AR's in 223 and like the Varmint Grenade bullets, or some of the other good varmint bullets. Pelts are not an issue here so the size hole you make doesn't matter. I have used everything from a .14 caliber to 12 gauge in all type guns and like .44 or .45 caliber revolvers best. This time of year I carry an old Single Six with a 22 mag cylinder and it has killed several over the years. The young ones can get out and be dumb enough to offer a shot in the spring and summer. I seldom hunt coyotes but never turn down taking a shot when the chance comes open. For open country with a rifle a 243 in a Ruger #1 has reached out and worked well when a friend and I hunted some. Probably should get the old rifle out and shoot something with it.
Only one I've ever taken was with a Winchester '94 in .30-30 when she wandered out in front of me while I was deer hunting. The house is in the woods, with a field behind, so I keep rounds near my Interarms Mk X in .270 if one hangs out around the house. Haven't had a chance to take a shot with it, though, so I guess I don't have much experience to offer. I also don't hunt them intentionally.
Coyote encounters are typically nighttime around my place. They get out and stalk my dogs or the barn cats. I like to carry my Remington 597 Archangel kitted (AR15 type) rifle with laser and red dot zero'd for 50 yds. You have to be quiet but the 22LR is perfectly capable of discouraging these critters and if there is a miss, you are less likely of shooting through the brush cover and hitting something unintended so I don't haul out the AR15 or SU16 or my 30-06 often as there are houses that can be hit in the distance (about 1/4 mile to half mile away). IF there are other things coming around the property at night (you can tell the difference in the way the dogs bark), I carry something heavier, like the SU-16 or the AR15 with hi-cap mag and red dot optics.
Tikka Whitetail, .270 Win. 130 gr. Scirrocco's. 4x12 Leupold Vx2. Harris bi-pod. 3.5" high @100yds. Dead on @300. -11"@400. The last 4 song dogs I shot were at 149 yds, 21 yds. 381 yds. and 440 yds. 4 with 4 shots.
I've only ever shot coyotes while deer hunting, all with a .308 M7 Remington stainless. Oh, I have shot one with a .257 Roberts M722 short action. I don't really hunt coyotes. These were just targets of opportunity. I'm a meat hunter. Unlike our president, I don't eat dogs.
Some of us were talking about this today after reading the OP. Most coyotes were shot while deer hunting or hunting other game. That puts deer guns high on the list of guns used by a lot of us. Also most were targets of opportunity so it was whatever gun that was handy. Favorite shot is through the lungs broadside. They may run but fold up quick when they run out of air. Lot of good stories about coyotes, as a friend said more fun than shooting s deer, don't have to clean it.
My favorite recent shot was like this : I was set up high on a ridge and called him in. He stopped at 149 yds. and sat down on his haunches!. So it was a straight on frontal shot and hit im square in the breastbone as he sat facing me. My .270 win, loaded with a 130 gr. Scirocco @ 3180 fps. literally lifted him straight up in the air about 4 -5 ft. off the ground, flipped him backwards like a gymnast and he landed about 7-8 ft. behind where he was sitting. I wish I could have had it on film. Could help but laugh outloud. He never knew it was coming.
One reason I bought my AK-74 earlier this year rather than the AK-47 is that I've heard
good things from others about what a good round the 5.45x39 is for coyotes. I haven't
had a chance to go coyote hunting yet, but if and when I do I'm bringing the AK-74 rather
than the deer rifles I own now.
The .14 is a sub caliber, meaning smaller than a .17 caliber. Sort of like James Calhoon's .19 caliber stuff, all wildcat,custom, and fun. Nothing you will find in Wally World, back to the topic of coyotes and guns.
Haven't hunted yotes in years. But when I lived in Arizona I would let my 2 Akita/Wolf mix dogs handle stirring them up and killing them. My female would get them to chase her then I would release my male to run in for the ambush. Kinda like a role reversal of sorts. I sure do miss those 2 dogs. They both died in 93 of kidney issues which both Akitas and the Mexican Grey Wolf are succesceptible too. I did carry my .30-.30 just in case. Sure miss that gun too.
EDIT: They died in 03 not 93. In 93 I was shit deep in the Middle East somewheres.