My #1 recommendation (especially if you travel on weekends or holidays) is to book well in advance. The less flexible you are on travel dates, the earlier you should buy. I mostly fly on Chase Ultimate rewards points, which function like cash on any airline... but whatever airline you fly it is always best to CALL to book. Do not search using the online rewards portals, they suck. There are less options and they do not let you plan creative things like stopovers. You'll find the best bang for your buck using these tools, noting the flight numbers and then CALLING to book it manually.
Google Flights: Explore Destinations
If you don't know where you want to go, Google Flights will show you fares across the world for your specified date range.
ITA Matrix: The Source Code
Now we're getting serious. Let's exhaust all flight options using the system that all of these portals connect to: ITA Matrix. Learn How to Use ITA Matrix in this article. ITA Matrix allows you to search stopovers, open-jaws, hundreds of airports across a full calendar. Worth noting, you can only search flights on ITA, you cannot execute the buy. Find the exact flight you want and then call Ultimate Rewards to book. The benefits of ITA Matrix include:
Searching an entire calendar month for lowest fares (not just plus or minus 3 days)
Seeing flights and connections visually in "time bars"
Finding airports within 25-2,000 mile radius of your airport
Stopovers and open-jaws using advanced routing code "X / minconnect 1440"
For example, a minimum connection of 24 hours in London Heathrow is "LHR / minconnect 1440" which is displayed in time bars below.
Daily Deal Aggregators: Incredible Fares, Often Weekday Dates
The sites below scan the market for low-cost (or even mistake) fares and can be a great source to check if you are flexible in your travel and live in a major hub. You can even take advantage of a major deal to cross an ocean, then change to a budget regional airline once you are there. It is better to browse the deal listings from your city rather than searching for specific itinerary through these tools. They are more valuable as deal aggregators than a search engines. Here are ones I use!
Finally, here are a few sites to explore that list regional budget carriers. Always good to compare a low-cost flight to a main hub like Munich, Hong Kong, etc. and then transfer to a regional airline.
Europe: SkyScanner lists many of the discount European carriers (examples below). It also let's you put "everywhere" in the search bar (much like Google) and see the fares in ascending order.
Often budget carriers do not pay the listings fee to be included on these search engines, so be sure to read the Wikipedia article of your destination airport to see all carriers that service that airport. Southwest, for example, does not come up and you must check the site individually. Check Wikipedia to see if there are any budget airlines that do not come up in your airfare search engines.
Hopper: Real-Time Notifications
Most airfare websites (including Kayak and Airfare Watchdog) have fare alerts emails, but in the world of dynamic pricing (where prices change hourly) the lowest fares are often gone by the time the daily email comes. Hopper will notify you in real time when the flight reaches its lowest point. When it displays a notification on your phone that fares have dropped, "hop" on it!