It Depends on the Material
There are three common egg-carton materials. Molded paper-pulp (the gray/brown fiber kind) is widely accepted in curbside recycling and is also compostable. Clear plastic cartons are usually PET #1 and are recyclable only where your program accepts #1 containers. Foam/polystyrene (#6) cartons are rarely accepted curbside because expanded polystyrene is hard to process — those go in the trash or to a special foam drop-off. Whatever the type, empty it and wipe out any egg residue first.
Paper-Pulp Cartons: Recycle or Compost
Molded-pulp egg cartons are the most versatile. Put them in curbside paper recycling, or tear them up and add them to a compost pile — they're a 100% compostable "brown" (carbon-rich) material that typically breaks down in about two to four weeks. They're also great reused for starting seeds in the garden: fill the cups with soil, plant, and transplant the whole cup. Local farmers, chicken keepers and schools often welcome clean cartons too.
Plastic & Foam Cartons
Clear PET #1 plastic cartons can be recycled in communities that accept #1 containers — but acceptance depends on your local processor, so check first. Foam (#6) cartons are the problem child: most curbside programs reject them, so reuse them if you can, or put them in the trash (or a foam drop-off if one exists near you). Foam and plastic cartons are not compostable.