There is a whole lot of navel gazing going on in The Heart is a Lonely Hunter. In fact, many of the novel's characters are downright self-absorbed. But all that self-involvement comes from a good place. They are just trying to suss out who they are. This is way easier said than done. If a character seems relatively sure who he is on the inside, he is woefully unsure who he is to other people. And if a character seems confident in her relationships, it's clear she has been ignoring the kind of self-exploration that comes with finding one's identity.
Singer's identity is entirely constructed by those around him. We never understand the real Singer, and neither does the man himself.
Singer is the only character in the novel with a true sense of self. Because he is deaf and mute, he doesn't have to put on a show for others; he just gets to be himself.