The Haynes Concorde manual says of the Tu-144 that "After 55 flights, reliability issues forced a withdrawal from service."
It also states:
"So much had been achieved; it may only have been the final few aerodynamic and control refinements that eluded the Tupolev dynasty."
Which might well be true, aside from an airframe construction method which turned out to result in a horribly dangerous structure.
The Tu-144 Wikipedia page states:
"A serious problem was discovered when two Tu-144S airframes suffered structural failures during laboratory testing just prior to the Tu-144 entering passenger service."
"it turned out that large whole-moulded and machined parts contained defects in the alloy's structure that caused cracking at stress levels below that which the part was supposed to withstand. Once a crack started to develop, it spread quickly for many metres, with no crack-arresting design feature to stop it. In 1976, during repeat-load and static testing at TsAGI (Russia's Central Aerohydrodynamic Institute), a Tu-144S airframe cracked at 70% of expected flight stress with cracks running many metres in both directions from their origin."
"While fatigue cracks of an acceptable length are normal in aircraft, they are usually found during routine inspections or stopped at a crack-arresting feature. Aircraft fly with acceptable cracks until they are repaired. The Tu-144 design was the opposite of standard practice, allowing a higher incidence of defects in the alloy structure, leading to crack formation and propagation to many metres."
And then there were the day-to-day reliability issues, again from Wikipedia:
"Early flights in scheduled service indicated the Tu-144S was extremely unreliable. During 102 flights and 181 hours of freight and passenger flight time, the Tu-144S suffered more than 226 failures, 80 of them in flight. (The list was included in the Tu-144 service record provided by the USSR to British Aircraft Corporation-Aérospatiale in late 1978, when requesting Western technological aid with the Tu-144, and probably incomplete.) A total of 80 of these failures were serious enough to cancel or delay the flight."
"Failures included decompression of the cabin in flight on 27 December 1977, and engine-exhaust duct overheating causing the flight to be aborted and returned to the takeoff airport on 14 March 1978"
"On 31 August 1980, Tu-144D (77113) suffered an uncontained compressor disc failure in supersonic flight which damaged part of the airframe structure and systems. The crew was able to perform an emergency landing at Engels-2 strategic bomber base. On 12 November 1981, a Tu-144D's RD-36-51 engine was destroyed during bench tests, leading to a temporary suspension of all Tu-144D flights. One of the Tu-144Ds (77114, aka aircraft 101) suffered a crack across the bottom panel of its wing"
"Tu-144 pilot Aleksandr Larin remembers a troublesome flight around 25 January 1978. The flight with passengers suffered the failure of 22 to 24 onboard systems. Seven to eight systems failed before takeoff, but given the large number of foreign TV and radio journalists and also other foreign notables aboard the flight, it was decided to proceed with the flight to avoid the embarrassment of cancellation."
"The final passenger flight of Tu-144 on around 30 May 1978 involved valve failure on one of the fuel tanks."
"Soviet decision-makers had little confidence in the Tu-144 when passenger service began in 1977. Considering the high rate of technical failures their reasoning was sound. Bookings were limited to 70–80 passengers or fewer a flight, falling well below both the Tu-144's seating capacity and the demand for seats. On its 55 scheduled flights, Tu-144s transported 3,194 passengers, an average of 58 passengers per flight. With officials acutely aware of the aircraft's poor reliability and fearful of possible crashes, Soviet decision-makers deliberately limited flight frequency to as few as would allow them to claim to be offering a regular service, and they also limited passenger load to minimize the impact and political fallout of a possible crash."
"A problem for passengers was the very high level of noise inside the cabin. The noise came from the engines and the air conditioning. In addition the unique active heat insulation system, which used a flow of spent cabin air, was described as excessively noisy. Passengers seated next to each other could have a conversation only with difficulty, and those seated two seats apart could not hear each other even when screaming and had to pass hand-written notes instead. Noise in the back of the aircraft was unbearable."