Yeah...good luck with that , Josh. Targeting wild canines but avoiding domestic dogs is a challenge at times.
Here are some things I do that may have helped me a bit over the years...
1.) I live in eastern Nebraska, where pheasant hunting is big, and with bird hunting comes a lot of dog activity. I specifically make coyote, fox, or cat sets in areas where there tends to be very little cover or bird hunting activity. My favorites: flat-picked bean fields, mowed hay fields, low-grazed pastures (no brush, very little cover, maybe a stock pond or two in the section), and then any harvested corn field edge near a bean field or pasture. I also like fencerows that run through any of the above, especially if the fencerow has only a little cover...it still tends to be a natural travel route and the most prominent eye focal point in the area at times. Find an intersection of two or more fences and that is also a good location.
2.) hoof or drive it a little further back into the sections, away from the farm houses. That is a challenge around here, lots of acreages and farmsteads. The farther I go off the beaten path, the fewer domestics I tend to catch. it takes a little more time away from checking other traps if you have to drive or walk in a little further...but so what. if you walk in, as I do, you also get to take another notch or two off your beltline during the season and that is never a bad thing. If I set along tall CRP grass, I move away from the edge of the grasses at least 15' or more and make a mound-type set. Has a little eye-appeal, but tends to miss bird dogs who work the field more. I also like little hill-top islands, maybe a little stand of trees on top of a hill all out by itself...often not worth the walk for a bird hunter across the picked field, but very attactrive to late-night hunting canines. Its not dog-proof by any means, but has reduced some catches for me in two of my locations I trap.
3.) make a few blank sets with traps near heavily baited/lured sets with no trap. Sometimes you can play on the wild canine's instincts to check out the blank dirthole with its only attraction of fresh dirt and the hole for eye-appeal. Sometimes that has no interest to a dog, but its not foolproof.
4.) if you do accidentally catch a dog, just remake that same set and leave it. Dogs learn fast. the dog scent is appealing to coyotes, I have found. Must be a territorial thing. I always remake those sets and have not been dissappointed yet. Sometimes I even use dog droppings at other sets...of course only in areas with no other dog activity, but it does interest coyotes.
5.) I get a lot of flack for this at times, but I actually scaled my trap size back and focused more on placement and a little guiding...I don't use anything bigger than a #2 long now. It has the same jawspread as a 1.5 coil, but with two long springs is a tenacious holder. I get away with it here because I am convinced we have smaller coyotes. A #25 or #30 male would be a trophy for me, and I have caught a lot of coyotes in my 25+ years trapping this area. A #2 coil actually has a larger jawspread than a #2 long, also. I use a lot of those little #2 longs and they have not let me down yet either. I have beefed the frame up on them and added boosters only for speed. I probably couldn't get away with them out west or in other areas, but they work for me here. I also have a lot of cover, lot of population density, and species mix. Any dirthole I set can produce coons, fox, coyote, cat, whatever...very little open areas around here. I downsized a bit to be easier on the other catches also. Anyway, sometimes a really big dog with a big foot gets missed in my little #2s, so I just went with this system.
6.) I haven't found a bait or lure that works on coyotes but not dogs. Fish is fairly close, and maybe some fruit-based stuff. Coyotes love fish around here.
7.) blind sets are an OK option also. find a trail, find a pinch point or good place and make a blind set.
8.) if you snare, be sure to use a relaxing lock and no kill poles. dogs that are used to being tied up have the advantage of knowing not to pull hard against the restraint. not every dog is used to that, though. be careful with snare use.
9.) log crossovers on small creeks have worked well for me, too. I snared a bunch of coyotes off of these and no dogs yet, to date. I like these during higher water times...which unfortunately has not been the case in recent years. When its cold out, the log crossing is the easiest option. Make a good trail down to the log and up the other side if it doesn't already have one. Same with beaver dams...I snare a lot of coyotes off of these on smaller creeks. Maybe they are used to using them to cross whereas a dog may not have the experience yet.
10.) the worst dog is the roamer who runs after hours, farm dogs who chase deer and run the countryside after the owners go to bed or leave for the evening. they aren't afraid of you, and are used to everything in nature. I have the most problem with these individual animals. i had one that I caught nearly 2 miles from the farm where he supposedly belonged. next day I had him again 1.5 miles in the other direction. I drove him home both times as I knew the dog and its owner. What do you do with a situation like that? Not much. Just be courteous, carefull with someone else's pet, and keep the PR positive. I think the smaller traps helped me here also by keeping damage to nothing. I watered the dog both times and gave him half my sandwich I had on me for lunch. We were pretty good friends by end of that season. I think he saw me as more of his hero for saving him when I released him vs. the one who made the set in the first place. Oh well...that's dogs for you.
Have fun!!!