Birds are generally much easier, since they'll sit out of reach of the cat, but very visible. I've just done the equivalent of catching a spider, only using a bucket and picture frame.
With small rodents, it is much more "fun", and I've generally had to rely on Zev to find whoever she's brought in, and then grab it before she does. In the past I've come to the conclusion that the mouse has escaped, only to find Zev, some hours later, going nuts in some part of the house, having cornered it.