There are laws governing the harassment, trapping, relocation and removal of vertebrate animals. Exceptions are rodents in your home or animals committing crop damage on your farm. You still may need to check control measures out with local warden.
Most avian damage is diurnal and can be observed directly.
More difficult because much animal damage is done at night. When ingested bird flies erratically making distress call then dies. Specific designs for different species. Same methods as discussed in capture chapter. For predators, leg hold, conibear and snare traps.