Is it dangerous for a supersonic aircraft to fly at exactly 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.