Here are some suggestions:
Cheapest solution: so you don't have to amplify your sound, consider HearFones. These gadgets are like having a PA system that requires no electricity or batteries. Wearing them will also have the added benefit of keeping you from using too much air pressure. You won't like yelling at yourself:)
Try garnering a little good will by contacting and alerting neighbors about when you intend to practice, being willing to work around times they especially need quiet (working night shift and need to sleep, baby's napping, etc). Your neighbors could become supportive friend-fans!
For an apartment, hotel room or space you don't own, you can try warming up and singing in the shower! You can also try vocalizing into a pillow or window curtains. However, don't practice in 'guarded stance'. A hunched over, careful posture and numb delivery could lead you into bad vocal technique, causing you to tighten up instead of loosen up!
You can always practice in your car... but if you practice correctly you must be mindful of how you are singing. SO.... don't tailgate, stay away from other cars, or better yet... park and sing!
For your home, you might consider dedicating a space for playing/listening to/practicing music. In my home, my music room is located over my garage, and there is carpet on the floor. It really is a great situation for me and my clients.
If you want to go a little (or a lot) farther with it,
- Get some soundproofing or sound absorption materials for the room. While there are cheap ways to do this, don't line walls with igniteable material.
- Construct a vocal booth
- Buy a vocal booth already made.
- Professionally design/build a soundproof room from scratch... or better yet, have an acoustical engineer or expert in architectural acoustics design it and build it for you.