Keep them flat, in the packaging they came in, somewhere cool, dry, and out of direct sunlight. An ordinary room indoors is ideal. Stored like that, stickers and labels keep well. Heat, humidity, and pressure are what age them before they're ever used.

What to avoid

Heat. A hot car, a sunny windowsill, or a shelf above a radiator softens adhesive, which can start to creep at the edges and make sheets stick to each other.
Humidity. Damp storage, like a garage or a basement, can make paper labels curl and paper backing wrinkle.
Pressure. Heavy boxes stacked on top flatten rolls out of round and press sheets together. A roll that's out of round can feed badly through a label applicator.
Direct sunlight. It fades print even before the sticker goes on.

Sheets and single stickers

Store them flat, in their packaging or a folder, rather than standing on edge, where they bow. A curled sheet of stickers flattens again if you leave it under something light overnight.

Rolls

Keep rolls in their packaging, and don't stack heavy things on them. If you use part of a roll, put the rest back in its packaging so the edges don't collect dust. A label bonds to dust on its adhesive edge instead of to your product.

Use them while they're fresh

Adhesive changes slowly with age, especially in warm storage. Order for the season or the batch rather than for years ahead. Before a big labeling session with old stock, try a few on the real container first.

Before you apply

Let stickers and labels reach room temperature before you use them, and apply them to a container at room temperature too. See applying stickers and labels in the cold.