Which aircraft can fly at FL600 or above?
Commercial Airliners:

Both Concorde (service ceiling FL600) and the Tupolev Tu-144 (service ceiling FL660) could reach FL600. Most airliners have a service ceiling of around FL410 though.
Business Jets:

According to this article the highest service ceiling for various business jets is at FL510, so none can reach FL600.
Military Reconnaissance Aircraft:

The Lockheed SR-71 Blackbird had a service ceiling of FL850 and routinely operated above FL600. Many other aircraft in this category (e.g. MiG-25, Tsybin RSR, U-2, RB-57F, M-17 and M-55) can also fly above FL600.
Military Fighter Jets:

Most fighter jets have a service ceiling below FL600, but there are a few that can fly higher (e.g. F-15 and F-22 with service ceilings of FL650, the MiG-25 again, and  MiG-31 with the highest service ceiling of FL820+ for a fighter).
Experimental Aircraft:

Wikipedia has a list of altitude records, often by experimental craft. The 60,000 feet limit was surpassed around the end of the second world war with the advent of rocket and turbojet engines. Two notable achievements:

The rocket powered X-15 reached space twice. In 1963 it climbed to 67 miles, nominally corresponding to FL3530. Note that the X-15 did not self-start.

The glider (!) Perlan II has a service ceiling of FL900 and actually reached 76,124 feet. It utilizes standing waves forming at mountain ranges which under certain conditions can extend into the stratosphere. From wikipedia: "A sailplane at 90,000 feet altitude flies in approximately the same aerodynamic regime [...] to be experienced by a moderate size aircraft flying near the surface of Mars."


High altitude balloons and rockets can of course fly higher.