There are pictures in the predator forum where predators have torn chicken wire, usually dogs but sometimes other things. There are photos in the predator section where predators have torn hardware cloth, again usually dogs.
Any wire you use will give you some protection. There are different things to consider with the wire itself. One is the gauge of the wire. The smaller (heavier) the gauge the more protection it gives you. Another is the size of the holes in the wire. Pat of that is that certain things can get through but also the bigger the holes the easier for a critter to get a grip so it can really pull. Another is the way the wire is put together. Chicken wire is held together by twisting, lighter gauge may be able to be untwisted by a strong predator pulling on it. Hardware cloth is welded. It’s possible in the light gauge hardware cloth for the weld to be broken. Chicken wire and hardware cloth come in different gauges and with different sized holes. The heavier the gauge and smaller the holes the more expensive it is. Typically hardware cloth is better for predator protection in several aspects but how much depends on the respective fencing.
Another big issue is how the fencing is attached. Your attachments can easily be your weak point. One, whatever you use to attach it might not be strong enough or get enough of a grip. Two, you may have edges where a critter can get a good grip to pull. What you are attaching it to could be the weak point, either letting the staple, nail, or screw pull out, or breaking or splitting itself.
It’s not just what materials you use, fencing, attachment methods, and what it is attaching to. It’s also your construction techniques.
Even in suburbia you may be amazed at which predators you have in addition to raccoons. Did you know that over 5,000 coyotes are estimated to live within the city limits of Chicago? Coyotes are big enough to tear some chicken wire. You could easily have bobcats and foxes. Skunks and possum are almost certain. Chicken wire will give you some protection from a lot of these but hardware cloth will give you more, gauges being equal. In suburbia dogs are probably your biggest risk, depending on how secure you are from dogs getting in that area to start with.
The way I went about it was to use 2” x 4” welded wire for my run and lined the bottom 18” with chicken wire. It’s not perfect but the 2x4 wire will stop anything up to but not including a bear and the smaller holes in the chicken wire stops many others. Where I attached it to heavy wooden posts I used 1-1/4” fencing staples, those ¾” poultry staples just don’t cut it. Where I attached it to a flat wooden surface I used furring strips. ¾” thick boards laid over the ends of the wire and attached with screws. The screws go through the holes in the wire. If the wood is soft I use fender washers over the screws to keep the screw head from sinking in too far. I drill pilot holes so the screws go in a lot easier and they don’t split the wood. By clamping down tightly that wire isn’t going anywhere plus you get the advantage of covering the edges so a critter cannot get a good grip to pull. That also keeps me from snagging skin or clothing on the fencing edges.
You don’t have to do it my way, there are plenty of other ways out there. The more secure you make the run the more it will probably cost you. It’s not always an easy answer. Good luck!