There is a whole list of bird to chose from...
#1. Canaries should be highest on your list due to the fact that they are AMAZING singers and do not require a complex diet.
http://www.parrotparrot.com/species-guides/canaries/
"Song Canaries: Perhaps the most famous canaries of all, these are bred strictly for their song. Most are yellow, green, or variegated. Best known are the Rollers, Waterslagers, American Singers, and Timbrados."
"Noise: Some of the type canaries can sing loud, rather shrill songs, but they are the exception to the rule. In general, the canary is one of the sweetest-voiced birds on the planet. Song canaries are trained to sing in harmony with others of their kind, and their owners compete vigorously for the prestigious prizes awarded their proteges. You can find quiet-singing canaries (Rollers), moderate-singing canaries (Waterslagers and American Singers), and loud singers (Timbrados, Type, and Colour Canaries). Rarely will canary owners receive complaints about the noise of their birds, which makes them particularly ideal for apartment dwellers."
#2. Doves not sure if you'd call it a 'song' but they have lovely cooing sounds...they also make a laughing sound that makes me laugh all the time! These also do not require a complex diet. there are diamond doves and ringneck doves...the more exotic ones like fruit doves will require more complex diets.
There are some incredible song birds out there like pekin or magpie robins, shamas, asian song birds, red wiskered bul bul, Hwamei, ect...Mynah birds and starlings can talk amazingly well....
but these usually require a more demanding diet and usually are fairly expensive.
actually if you'd like you could even keep both doves and canaries in one encloser...
http://www.birdchannel.com/bird-species/find-the-right-bird/doves-and-finches.aspx
http://softbillsforsale.com/