Make some starling traps.
Essentially, you start with a soup can, and a wood box wide enough to put the can between the sides.
Operations principle is based on the Starlings and sparrows desire to enter a hole to nest or shelter. They enter the hole, the weight of the bird causes the can to drop, they exit the other side of the box into the holding area, the trap resets itself.
Here's one version.
http://www.modernfalconry.com/sstrap.html
You can make it up to look like a birdhouse. Some guys use PVC plumbing pipe to mount the trap as well as to feed the catch into the hold.
A 20L pail makes a good hold. Melt some screen into part of the lid, so the birds can breathe, and leave an apple or two inside for food and water.
Get a bird book, and a spray can of starting fluid (ether). The bird book is to ID the songbird species, so you can pick them out. It's illegal to kill songbirds. Pay close attention to the sparrows you catch. Some are protected species. Most will be English Sparrows, which are not.
Anyway, a squirt of ether and cover the vents on the bucket, and you are done with the days haul of pest species.
Do take down your trap if you are not able to check it at least daily, eh.
Do your best to birdproof your house, too. All the traps you can build, won't stop the skyrats from coming in from the rest of the city. Best you can do is to thin the herd.
Cheers
Trev