Re carnet worldwide - a useful thread
Below is a useful HU thread. It's from a few years back, but the map showing the countries for which a carnet is mandatory and those for which it is recommended is a useful guide.
Syria is a good example. It is recommended, but not mandatory, as the replies to the previous thread suggest. I've just left the region and had a carnet since it was part of longer trip East. It was stamped in Syria, Lebanon and Jordan.
All the best,