Is there a list of game that can be hunted year round like sparrows, starlings, chipmunks etc...? Also, can rabbits be shot if they are destroying your garden?

This link will take you to all the game species seasons: http://www.ct.gov/deep/cwp/view.asp?a=2700&q=534710&deepNav_GID=1633

Other than that, here are the non-regulated species:

Sec. 26-92. Wild birds other than game birds protected, exception. Game birds defined. No person shall catch, kill or purchase or attempt to catch, kill or purchase, sell, offer or expose for sale or have in possession, living or dead, any wild bird other than a game bird, or purchase or attempt to purchase, sell, offer or expose for sale or have in possession any part of any such bird or of the plumage thereof except as acquired under the provisions of this chapter. For the purposes of this section, the following shall be considered game birds: The anatidae, or waterfowl, including brant, wild ducks and geese; the rallidae, or rails, including coots, gallinules and sora and other rails; the limicolae, or shore birds, including snipe and woodcock; the gallinae, including wild turkeys, grouse, prairie chickens, pheasants, partridge and quail; the corvidae, including crows. No person shall take or destroy any nest or any egg of any wild bird or game bird. No person shall possess any nest or egg of any wild or game bird. English sparrows, starlings and, when found depredating ornamental trees, agriculture crops, livestock or wildlife, or when concentrated in such numbers as to constitute a public health or public safety hazard, crows, rock doves, monk parakeets and brown-headed cowbirds shall not be included among the birds protected by this section. Any conservation officer and any other officer having authority to serve criminal process shall have the same powers relating to violations of the provisions of this section as are conferred by section 26-6.

Chipmunks are not protected, neither are rats, mice, moles.

Because there is a specific season for rabbits, you could only live trap them out of season and relocate them away from your garden.

...or shoot them once.