Veneer is thin--usually only 1/32 of an inch--so it doesn't need an industrial adhesive to stay put. But it does need an adhesive that dries hard. Any one of the options below is a good choice to use to glue your veneering project. The important thing is that you completely cover the area of the base wood the veneer is to be applied to. This assures a lasting bond with little chance of moisture seeping in to compromise it.
Use carpenter's glue. This type of glue is available in home improvement stores everywhere. Spread a thin layer on the piece to be veneered. Lay the veneer over the glue and position carefully. Clamp veneer to base piece until glue dries. Follow manufacturer's recommendations for drying and set-up times.
Use iron-on veneer glue. Apply a thin layer to both the base and veneer piece and let dry. Position veneer on base piece. Iron with a regular clothes iron set to the dry setting. Follow manufacturer's instructions for best results. This works best on small, curved or oddly shaped pieces.
Use cold-press glue. Cold press glue is made for veneering. It is much thicker than regular carpenter's glue. It comes premixed in three wood tones--light, medium and extra-dark--and leaves a less-conspicuous mark from the glue. Spread cold-press glue over the base piece. Apply veneer and clamp until glue is dried and set up. Follow the manufacturer's recommended times for glue to set up and dry.
Use contact cement. Now available in a water-based version, contact cement is for use with paper- or wood-backed veneers only. Spread the contact cement on both the base and the back of the veneer. When the glue dries to a tacky feel, the pieces are joined together. You only have one chance with contact cement; "contact" means it binds on contact, no exceptions.
Use polyurethane glue. Polyurethane glue bonds nearly any surface. This is a good choice when veneering a non-porous surface, such as plastic or metal.