Scientists studying the songs of songbirds have found it to be mostly a learned behavior subject to environmental influences like rearing and food availability. In fact, researchers from the Max Planck Institute for Ornithology in Seewiesen compared song and brain structure of parents and offspring in zebra finches that were raised with genetic or foster parents. They also varied the amount of food during breeding. They found that song and underlying brain structure didn't appear to be as affected by genetics as by strong environmental factors.
These results related to a human behavioral biology question (i.e., which aspects of behavior are learned compared to those expressed by genetic predisposition). It's known that human personality and behavior are much less set by genetic background. Environmental factors shape brain and behavior by so-called epigenetic effects. In fact, hormones play an big role and can have long lasting effects.
Research with zebra finch breeding pairs (where half of the hatchlings were raised by their genetic parents, and the other half were raised by their foster parents) showed this too. At 100 days when the male offspring were grown, the researchers recorded their songs. The results showed that genetic heritability was low for most song characteristics, except the number of song syllables and maximum frequency. The rearing environment and song of the foster father mainly predicted the unique syllables of the sons' songs and was dependent on food availability.
So keep feeding the birds in your backyard and providing a healthy environment. Songbirds will be happy and singing. Go Science!