No matter what you do, you will inevitably find yourself battling air bubbles at some point when making découpage art. As the paste and paper dry, a bit of shrinking occurs when the moisture evaporates, which can cause bumps and bubbles, even if the découpage was perfectly flat when wet. Though air bubbles are not entirely avoidable, they can be minimised and sometimes removed.
Prep the Surface
You can découpage both porous surfaces such as wood and nonporous surfaces such as glass. In either case, the surface must be completely clean so the paste will adhere properly, minimising air bubbles. Even if your object appears clean, wash it or wipe it with a damp cloth. When applying découpage to wood, sanding the surface first can help the paper adhere -- just be sure to thoroughly wipe away all of the resulting dust before continuing.
Don't Use Too Much Paste
Using too much découpage paste will cause problems with bubbles and is more likely to cause the paper to buckle. Apply the thinnest layer you can on both sides of the paper. If you notice tiny bubbles appearing on your previously-perfect découpage as it dries, they can be remedied with a quick prick of a pin.
Burnish the Paper
Burnishing can be as simple as pressing the paper flat with your fingers, or you can use a burnishing tool. Commercial burnishing tools are inexpensive pieces of plastic shaped like a thick letter opener with rounded edges rather than sharp ones. Rub the tool over the surface to push out any air bubbles. Without this step, expect bumpy découpage.
Watch the Humidity
Because découpage pastes dry by the evaporation of water, the more humid the day, the slower it will dry and set. This can be problematic, because the longer the paper stays wet, the more buckling can occur. Avoid decoupaging on humid days, or do it in a dry, air-conditioned setting. Decoupaging is not a good outdoor activity for several reasons, including the possible interference of wind and dirt; humidity is also a big reason to keep your découpage work space indoors.
Wait it Out
Although it may be tempting to fuss with your découpage constantly as it dries, many air pocket imperfections will work themselves out as the paste dries and sets. This is especially true if you've created more than one layer of découpage on the surface. Allow the piece to sit for a day or two to allow the paper to settle in place.