Fishing in the United States is regulated separately by each state, and coastal states generally have separate regulations for fresh-water and salt-water fishing. In most cases, spearfishing is not separately identified for special regulation, and you just need whatever fishing license would be required for line-fishing in the same waters. Spearfishing is legal in Mexico with a Mexican fishing license (see sample authorization letter. In Canada, fishing is regulated by Fisheries and Oceans Canada which in turn regulates fishing separately in six regions. Separate freshwater and saltwater licenses are required, but they are widely available at sporting goods stores and similar places.

Coastal State links:
California Sport Fishing Regulations
Oregon Hunting Licenses and Regulations
Washington Dept. of Fish and Wildlife
Alaska Dept. of Fish and Game Sport Fishing
Hawaii Division of Aquatic Resources Fishing Regulations
Fish Texas
Louisiana Dept. of Wildlife and Fisheries Fishing Regulations
Mississippi Dept. of Marine Resources
Alabama Marine Resources Division
Florida Fish and Wildlife Conservation Commission
Georgia Wildlife Resources Division
South Carolina Dept. of Natural Resources (Fishing)
North Carolina Wildlife Resources Commisssion (Fishing)
Virginia Marine Resources Commission (note especially spearfishing restrictions)
Maryland Dept. of Natural Resources Fisheries Service
New Jersey Bureau of Marine Fisheries
New York Bureau of Marine Resources
Connecticut Dept. of Environmnetal Protection Marine Fisheries Information
Rhode Island Fish and Wildlife Division
Massachusetts Dept. of Marine Fisheries
New Hampshire Fish and Game Dept.
Maine Dept. of Marine Resources Saltwater Recreational Fishing


In most states, spearfishing in freshwater is not legal. But note the following links for two inland states:

West Virginia Underwater Spearfishing and Diving
Utah Spearfishing