If a plane that is designed for supersonic flight, say the Concorde, kept flying at exactly the speed of sound, would there be any danger in that? If so, what could get dangerous at constant Mach 1 flight and how long can a plane survive in flying at Mach 1?
Make sure to read this answer to understand what is special about Mach 1.
Pilots regularly report that, approaching Mach 1, the airplane is shaken by oscillating shocks (they don't use that term, though, they speak of buffeting) and once the Mach meter needle crosses the 1, the airplane becomes calm and flight smooth again. Since the flow speed around the airplane is normally a bit higher than flight speed due to the displacement effect and the need to produce lift, supersonic pockets start to occur as early as at Mach 0.6 (depending on the particular design) and expand the closer flight speed comes to Mach 1.
Swept wings delay those Mach effects and the peak of the drag coefficient of supersonic-capable designs is also delayed to somewhere between Mach 1 and 1.2. While the drag coefficient normally drops a bit past this peak, drag itself normally rises with speed. Flight at Mach 1 is particularly inefficient but, apart from that, drag is not an issue.
What is more important is the shift in the center of pressure when changing from sub- to supersonic flight. It will move aft and requires the pilot to re-trim the airplane. Supersonic-capable designs normally use full-flying tails for that and delta airplanes raise their trailing edge flaps. Concorde added pumping fuel aft and the XB-70 reduced the rear part of the wing by folding its wing tips down. When flying at Mach 1, the airplane is smack in the middle of the largest pitch moment change over speed range, so any change in speed means also a strong change in pitch moment. When no trim change accompanies that change in speed, the airplane will be unstable in pitch: Flying faster will move the center of pressure aft and make the airplane dive (Mach tuck), speeding up more, and vice versa. Reducing speed might even lead to structural failure from the sudden pitch-up and increase in load factor if not corrected by control deflections.
That is what makes flight at Mach 1 unadvisable. If the pilot or the FCS keep pitch under control, flight at Mach 1 can continue until the fuel runs out and the airplane hits the ground, causing speed to drop dramatically.
Drag forces grow quickly as you approach the speed of sound, and then fall off somewhat after you are above the speed of sound. This means flying near the speed of sound puts you in a high-drag regime, where you are putting work into the flow as shock waves are trying to establish themselves at various points on the airframe. Unless your airframe is perfectly symmetric and smooth throughout, those shock waves can't be expected to be perfectly symmetric either, and the result will likely be quite a bit of buffeting and banging around- and really excessive fuel burn.
That's going to compromise the controllability of the plane as well as expose it to a barrage of shock loads that will eat up the fatigue lifetime of the structure, and furnish a rough ride for your passengers.
Mach buffet precedes mach, and begins in the transonic range. The transonic range does not begin or end at 1.0 M1. It begins typically around .85 and may continue to 1.2 to 1.5. Buffeting may occur at any point within that range.
Mach is not a point that is reached uniformly by the entire aircraft. This is to say, mach airflow will be reached at some points of the aircraft sooner than others. An aircraft may be at the lower end of the transonic range, while some portion of the aircraft is experiencing mach airflow. The 747, for example, experiences mach airflow at the rear of the upper deck supernal "hump" or dome. On freight 747's with little insulation, the airflow at the rear of the upper deck may be heard banging and snapping (this is not a "supersonic pop" or "bang," but rather airflow disruption and interference with surrounding airflow; it is a place where local airflow exceeds M1 in transonic flight).
In a sense, mach buffet can be thought of as the way a motorcycle feels on the highway, driving behind a big rig truck, or approaching the front of the truck, in the adjacent lane, on the highway. That shaking and buffeting from disturbed air. Get ahead of the truck, the buffeting goes away. Approaching the front of the truck, one begins to encounter air deflected from the front of the truck, and buffeting. Get just past it, and it smoothes out.
A big difference is that at highway speeds, the air isn't considered compressible. In flight, approaching mach, compressibility and shock waves become a factor; more than one occurs, both ahead of the wing and along the cord. When the shock waves have passed along the cord, the drag rise associated with approaching mach, and aft movement of the shock wave, are no longer a factor and the aircraft operates more smoothly. Other effects such as mach tuck, also associated with movement of the center of pressure stabilize.
The buffeting and mach effects are most prevalent in the transonic region.