The cheapest airlines may not be one of the “low fare” airlines. Those airlines (Spirit, Frontier, Allegiant) ticket prices are cheap because the tickets are “unbundled”. This means you’re only paying for the seat on the airline plane. Have a carry-on item? Extra $$. Want something to drink? Extra $$. Want a specific seat? Extra $$.
When you see their fare, it may look great, but it isn’t necessarily the best price. As a rule of thumb, add $50 for each carry-on and checked bag you’re taking, and if you’re not familiar with flying the airline, check their baggage fees. As soon as you add baggage, the low fare airlines might be comparable to Southwest and the full fare airlines.
When I’m planning my travels and see an “awesome” fare from one of the low-fare airlines, I automatically tack on $50 to account for flying out and flying back. Suddenly, they may not be one of the cheapest airlines.
International airline fares
European airlines are different than ours; there is rampant competition and their domestic airlines offer incredibly low fares. The cheapest airlines in Europe are RyanAir & EasyJet.
If you’re headed to mainland Europe, you won’t find a cheap fare direct unless you live on the east coast, and then it’s still a maybe.
Instead, search for fare from U.S. cities to Reykjavík, Iceland and Dublin, Ireland (2 cheap European cities to fly to), & then book RyanAir or EasyJet from there. The U.S. cities you should search are New York City, Boston, Chicago, Miami, Newark, & Sanford (they have a flight to Iceland).
It’s complicated, but it’ll get you a cheap fare. If this is too much trouble, you’ll pay more for easier searching.
Bottomline – Don’t buy a cheap airline ticket on a low-fare airline “just because”. Keep in mind baggage fees, etc…, and it’ll save you money.