It depends what you want to do. I don't have any experience staying in the same place for a long time (>2 days) and trying to find a free spot to sleep. For me, that's when i would just book into cheap accommodation or try find a short term rental. Personally i don't like the feeling of staying in the same place for extended periods without a secure shelter to leave my stuff. There are lots of people on this site with more experience living in a fixed location without any money who can provide better tips than me if that's what your goal is.
However, specifically in the context of traveling by bike, if you literally go to a new place every day... Then i think there are more options, because the local people will you see you more as an explorer or an adventurer than a bum, and they might be more inclined to help you, or at least not hassle you, because they know you'll be out of their hair 12 hours later.
One good place is to check out rural locations that are commonly used by people in the area for overnighting outdoors. Some states or provinces are better than others for this. Look for wildlife management areas, state or county fishing lakes, boat ramps, or any forests, parks, beaches, desert or greenspace that is government-owned. In some of these spots you can legally camp for free, other ones cost a trivial amount of money per night (<$10), or you might just need to get a permit. State and national parks have pretty great camping facilities, but tend to be a bit pricier.
Towns with fairgrounds sometimes have free camping at the fairground. In smaller towns (<1000 people) you will often find a handful of RV sites that you can pitch a tent at, in a pinch. Also in smaller towns you can talk to the locals and just ask if there is a good spot to camp. If you're obviously a traveler who is only going to be there overnight, then sometimes people will just point out a back yard, church ground or abandoned lot where you can set up. In one town i met the mayor in line at the general store and she said i should just pitch out back of an abandoned building. This is the sort of stuff you may not be able to get away with if you're staying in the same place for a long time, but if you're just passing through it's easier.
If you don't mind depending on other people's charity for a place to stay, then as a bike tourer you can also use the site WarmShowers, which is basically free homestays for bike tourers. More of the hosts tend to be in and around larger cities and popular tourist destinations than real small towns and backcountry, but if that's where you're headed anyways then i suppose it's not a big deal. Personally i stayed at a WarmShowers once and never want to do it again, i just felt extremely awkward having to rely on charity instead of being independent, but other people don't have that problem so it's up to you.
