There is only one truly international airport in Oman, and that is Muscat International Airport. That said, Qatar Airways offers direct flights between Doha and Salalah, and Fly Dubai offers direct flights between Dubai and Salalah.
Airlines that fly to Oman include:
British Airways (www.ba.com)
Fly Dubai (www.flydubai.com)
Gulf Air (www.gulfair.com)
Oman Air (www.omanair.com)
Qatar Airways (www.qatarairways.com)
Saudi Arabian Airlines (www.saudiairlines.com)