As in any other artistic field, singing lends itself perfectly to self-learning. You can learn to listen to your own voice and correct the notes that are out of tune, adjust the vocal cords and vocal timbre, master your breathing and, little by little, you can start calling yourself a singer. It's a lot like yoga or self-disciplined sports. You always have to develop patience, inner mental strength, breathing technique by listening to yourself, feeling like yourself and a good vocalist, we'll have to address all that even with a mentor.
I highly recommend the Berklee Vocal for Performance with Donna McElroy DVD, and then some practice, and then a little silence, and then listen, rinse and repeat. As a singing teacher who teaches singing classes for beginners, I have worked with many students who initially learned on their own.