Three awkward objects, three different setups, and one rule that covers all of them: daylight.
Bats
Long and varnished, which means reflections. Lay it flat in open shade, shoot along it rather than straight down from one end, and take the face, the back, the toe and the shoulders separately.
Signatures on a bat need light from the side so the ink lifts off the grain.
Shirts
Flat on a bed or a floor, square on, whole. Then the labels, the numbering, and the signature close. Fold lines are visible and should be — a shirt photographed to hide creasing reads as a shirt photographed to hide something.
Paper — programmes, tickets, scorecards
Flat, square on, even light, plus raking light for creases and foxing.
The shot that prevents the dispute
The signature, close, in neutral daylight. Warm indoor light makes faded ink look strong, and fading is a condition fact you will have to write down anyway.
Light
Open shade or daylight through a curtain. No flash on anything varnished, framed or glossy.
Provenance frames
Photograph the certificate with the issuer readable, and any photograph of the signing. Those are provenance; keep them separate from the condition shots. The ladder.
What goes wrong
Shooting a bat under a ceiling light so the varnish glares, cropping a certificate, and photographing a shirt folded in a bag.
Questions people ask
Should a signed bat be photographed standing?
Lying flat gives you even light along its length. Standing gives you a lean and a shadow.
How do I show a team-signed item?
Whole shot, then each signature group close, and say how many you can identify.
Video?
Yes — a slow pass along a bat shows the grain and the signature together.
More: describing it, common mistakes.