I’ve done two x-country self supported bike rides and I “urban camped” both times.
Firstly, are you staying anywhere near towns? If so your best bet is to go into town and start making friends. On my bike trips our group would roll into our planned town with no place to stay and anywhere between 30 minutes and 2-3 hours later have a spot. Either in a house, backyard, city park, behind a business - it really varied.
The easiest way is to find a good city park. Then find the cops or city hall and explain to them your trip and ask if you could camp there. This takes some word choice. Don’t say “Can I a camp in your park?” - better would be “I’m on this long distance unicycle trip, I was planning on staying in this town tonight and I was wondering if I could throw a small tent up in your park?” Avoid the word “camp” like the plague - it’s a killer. What also works is if the park has a shelter say “blah blah unicycle blah blah and I saw that nice shelter in your park - could I throw a sleeping bag under that tonight?” Also, always remember to introduce yourself with your first and last name and explain your trip before you ask for anything. Look them in the eye and shake their hand - you have nothing to hide.
What also works is going up to every church in town. “Hi, I’m (your name) and I’m (explain your trip) and I was wondering if I could throw a small tent up in your yard tonight.” If they are confused mention that you’ve had good luck with churches in the past.
If that strikes out go to the local library under the pretense of updating your website/blog. Strike up a conversation up with the librarian afterwards and tell her about your travels. Eventually ask if she knows anywhere that you could spend the night. She’ll probably offer hotels and campgrounds first - listen patiently and then follow that with “Yea, the issue is that I don’t have alot of money for this trip (since i’m saving up for school?) and I was hoping to just find a park or someone who’d let me throw my tent up in their backyard. I don’t need much space at all.” This will get his/her mind thinking about all their friends who may have yard space. They key with this kind of thing and with the supermarket trick that I’ll mention next is to BE PATIENT. Don’t hurry through a good conversation to get to the asking. People will think you’re just pretending to be nice to get something from them. Slow down and enjoy making friends, you’ll find a place to stay eventually.
If that doesn’t work go to the supermarket to resupply food - while there hopefully people will strike up converstations with you. Touring bikes and Unicycles are great coversation starters. Once you answer their questions and are super polite eventually ask “Do you know anywhere I might be able to throw a tent tonight? A park or maybe a backyard?” Emphasise how little space you need. Remember - you are screening them as much as they are screening you. Don’t stay with sketchy people.
As for getting into someone’s house - let them offer that. A lot of people will put you into their yard and then later invite you into their house for sleeping or dinner - but only after they’ve gotten to know you better first. Be friendly and engaging and once they are sure you aren’t a threat or a wierdo (usually at least a few hours later) they’ll start to really open up to you.
On rare occasions I’ve asked people who were out in their yards if I could camp there. That works ok, but is a last resort. People don’t want to be occosted in their private spaces. Public spaces are a much safer place to ask. The super-market trick works pretty well. Also, be prepared to strike-out a lot. Don’t be afraid of rejection. You’ll may be turned down 10-20 times before you get a good place, or get one first try. It depends on the town.
Small towns are better than big towns. Small towns are WAY better than big towns. 30k or smaller is what you are looking for here. I can find a place ten times faster in a town of 500 than I can in a town of 30 thousand. Big cities - don’t even bother unless they are hippie like Eugene or Missoula. Portland, maybe, but it would take all my skills and be pretty hard. (there I would start asking at co-op houses and natural food stores)
Finding a place to stay is one of the funnest parts of touring, and the people you meet will really enrich your experience. Do think about water availability at all times. If you camp in a park find a water spigot. If there is no water, then find somewhere or stalk up on gallon jugs before the store’s close. You’ll need water during the night, for breakfast, and to fill up before tomorrow’s ride. Also, if you are park camping let them know that you’ll be out very early in the morning so you don’t bother anyone. Don’t park camp without letting the cops know unless you are super well hidden. (like deep in the trees or in a baseball dugout.)
Trialside camping works too - but it’s sketchy because of the lack of available water, the chance of police intervention, and the chance of local hoolagins/territorial bums messing with your stuff. One guy with a bivy sack can probably find a place in any town to squat without asking permission so that is an option too - but not entirely my style since I kind of have fun asking. Many guys that I know use the “squat in a bivvy-sack” technique somewhere out of the way. If I were ever on a solo-ride I’d use that option more - but on my trips we had 4 people one time and 7 another so that wouldn’t have worked. We found a safe place to stay EVERY night, so I know you can do it.