Alright - everybody seems to be giving serious answers to this question - - So, I will respond again after some serious reflection.
The dogs temperment has more to do with it than the breed. I had a wonderful 85 lb pit bull who loved to chase cats. NOT EAT OR KILL - just chase. FOR Years she would chase a cat and when it stopped running she would sit and bark at it - - It was her favorite game. NONE of my neighbors were concern, she was harmless and they knew it.
As far as birds go, She never messed with LIVE BIRDS. Somebody in the neighboorhood had the big YARD ORNAMENTS - A FAMILY of DUCKS. One by ONE she brought them home to me. We drove around the neighborhood many times trying to find the owners so that we could replace Yard Ducks for them, but we could only find PINK FLAMINGOS in peoples's yards. She never touched those - - - only the ducks.
NOW, I don't want you to think we were irresponsible dog owners who let our dog run wild. We did not. When she was younger, she was a fantastic escape artist. First, we hot wired the bottom of her fence. Then, we discovered she was climbing - YES CLIMBING - over the fence. So we hot wired the top of the fence. Then we discovered that she tested the hot wire everyday. If the grass was damp from dew and tall enough to touch the bottom hot wire then it would short circuit the hot wire on LUCKY ESCAPE DAYS.
Finally, we called the vet and asked "WHAT DO WE DO?" The vet said she was suffering from separation Anxiety and we needed to get her a BUDDY or just learn to live with it. BY THIS TIME, the crazy dog was BREAKING into our own house. We would come home and find her laying on our bed. She would find a window that hadn't been fully shut the night before and climbed her 85 lb butt through it. AFTER that she just became a FULL TIME HOUSE DOG.
NO MORE MAJOR PROBLEMS. Occasionally on the weekend she would disappear while by other half and I were coming in and out, in and out, doing work around the house. But we would holler and she would come running home ( just like a child) as soon as we discovered that neither one of us had the dog! ! WHAT A TRIP - SHE WAS.
MORAL of the Story - - - BIG BAD PIT - - Not a KILLER. She did have the urge to chase though.
BIG DOGS - - St. Bernard comes to mind. This type of dog was never bred for hunting or herding. In general a very laid back dog.
SMALL DOGS - - Maltese - BUT they have lots of hair / grooming needs
I don't think you will be able to find the dog you want by looking at breed alone.
You will have to find a dog whos temperment does not include the urge to hunt or PLAY or CHASE moving living things.
You will have to find a dog who can be trained ( it won't come naturally) to leave the other animals alone EVEN WHEN YOU ARE NOT AROUND.
Even HERDING DOGS do their jobs by nipping at the live stock.