I'll be flying in the next few days and I'm really concerned about the potentially dangerous territories my flight might be crossing. In this case, I'll be travelling from Singapore to Switzerland and I'm worried about flying over Iraq and Syria. How can I find out what route my flight will take?
You can view the path for a specific flight on Flightradar24. Here's the flightpath for yesterday's SQ346 SIN-ZRH
If you wish to look up which flights there are from a given origin to a given destination you can use Flightstats. Note that any flight may have to change route for one reason or another, but you will be able to see from the history how likely this is.
Updated following Jan's helpful answer and David's comment. To see the Great Circle route that would typically be followed in the absence of any other air traffic or weather restrictions, you can enter the origin and destination airport on the Great Circle Mapper website and would see this route:
From the difference between the two images you can deduce the countries that the airline is particularly attempting to avoid or does not have permission to overfly.
Update: The FAA provides the following map of countries to avoid overflying: https://newrepublic.com/article/118764/map-faa-tells-airlines-avoid-flying-over-these-countries (may be out of date).
Flights will usually take a ‘curved’ route when viewed on a flat map with more horizontal streches in northern latitudes due to the ellipsoid shape of the Earth. That’s why the map Berwyn supplied also shows this ‘bent’ route rather than the seemingly direct straight line.
Also, flights typically do not cross areas with severe crises. In 2014, many airlines were already circumventing Ukraine. It became apparant why when the Malaysian Airlines aeroplane was shot down in the summer. That airline was actually one of the few still flying over the country in spite of safety concerns. So even if you are taking a route that might bring you across the areas in Syria/Iraq that IS controls, many carriers will purposely detour around these areas (and around the Sinai, in case you were wondering). Note how KLM flight Amsterdam–Singapore also detours around the Ukraine.
With the reputation Singapore Airlines has, they will likely be among the first to avoid critical areas.