Probably, I do with my 135. This is the place I got my hitch and they have one for the Civic http://www.etrailer.com/hitch-2013_Honda_Civic.htm
What kind of trailer? One of those junky single rails with the bike on it? Sure, why not. You see the Germans towing much more with much less.
If you are talking about one of those foldup trailers with just 2 wheels, a single rail in the middle and that is it. Probably. In general, you can tow just about anything with just about anything. Meaning, it doesn't take much effort to get something to roll. That is why you see strongman competitions with guys pulling 18-wheelers by hand with a rope, and a Toyota pulling the space shuttle. When you think about the physics behind it, it really isn't that impressive. The real question is, will you be able to stop it?
Definitely had some "ohfuckohfuckohfuck" moments towing my bike with a Malibu Shit. Guess I'm a Dave K and/or a wimp
civic should do just fine just be gentle, I helped my sister move back to illinois from florida pulling a 5x8 enclosed uhaul with her 07 civic lx(auto trans) as for bikes I cart around my bike on a 2 place open trailer with my prelude(manual trans) when I dont feel like using the truck
When you're talking about towing with a small car, start adding up the real weight of the trailer and bike, and then keep the car as empty as possible. Like Broome and someone else said, the towing part isn't as important as the stopping, and transmission wear. I think of it this way: How much would 4 obese passengers and their shit weigh? 4 people times 300 lbs = 1200 lbs. of human, plus another 50 lbs or so of luggage in the trunk for each person. That's only 1400 lbs. that the car is really made to stop, in addition to its own weight. The driver remains, so subtract your own weight from that 1400 lbs. That's how much you can tow, and safely stop... The smaller harbor freight trailers are about 300 lbs on their own. then 450 for the bike. Round to 500 to be safe, and you're at a solid 800 lbs. That leaves less than 400 lbs for additional stuff in the vehicle. Less would be better.
I bet the owners manual has a tow rating and I bet it's 1500lbs, seems like that's what most compact cars are.
Like the others say, with manual trans not a problem. I just mainly kept in 4th, dropped to third with the hills. Then again I was under the CA 55 mph limit, slow cruising at 60. Made this trip from LA to Sears Point for AFM.
Like most vehicles that tow. If you use all of the available tow rating, you will wear out the suspension, transmission and brakes much faster than normal. Also you will need to look far ahead and keep a bigger distance between you and the vehicle in front of you than normal. Even with trailer brakes, I can't stop as fast when towing as I can when I am not towing. Keep that in mind and you shouldn't have a problem towing with a Civic. Just make sure you keep up with or do extra transmission, suspension and brake maintenance and inspections. Also if it is just you in the car, put some heavy stuff that goes into the car (i.e. toolbox) into the front passenger floor area to balance out the weight a bit so it isn't all hanging over the rear axle.
I'd adjust that down to 1000. I just spent the last month or so researching what vehicle I wanted to replace my '02 Explorer, and I needed one that could easily tow a 2500-lb. boat. So I spent a lot of time looking at gross towing weight charts. From what I saw, the tow ratings mostly broke down into the 1000-3500-5000-7000 categories. Admittedly, I wasn't looking at compact cars but at mid-size SUVs and crossovers. I would expect vehicles like the Civic to be on the lower end of that scale. But to answer the OP's question: Yes. But lighter is better.