I'm going to focus on the evolution of the law itself, rather than its application.
After Stalin's death (1953), the worst abuses of the state were almost eliminated, but the process of expanding rule of law, judicial independence, and equality for the law took longer, way longer. The legal reform train sped up after Krushchev's denunciation of Stalin and his crimes at the 20th Party Congress in 1956. Some scholars say that Krushchev's "On the Cult of Personality and its Consequences", aside from denouncing Stalin, elevated the status of jurists and enhanced the role of law in Soviet society. The immediate time after Stalin's death and around the 1960's was a bit of an era of legal utopianism.
The first move was to abolish the system of extra-judicial terror. This went from eliminating the special boards of the Ministry of Internal Affairs, prosecuting all criminal cases (including political crimes) in the people's courts with regular judicial procedures, no more secret police allowed arrest without the authorization of a judge or procurator. The NKVD's leader, Beria, was even arrested for crimes against the state, and executed.
The Stalin era was one of extremely harsh penalties, which successors moved to reduce. Three weeks after Stalin's death, an amnesty was announced for many of the victims of Stalin's oppression. 1954 saw the reintroduction of parole, with adult criminals now being eligible for release after serving two-thirds of their sentences. Smaller economic offenses and less dangerous acts were decriminalized; the penalty for petty theft from state-owned factories went from 7-10 years in a corrective labor colony, to 3-6 months of corrective labor (sans confinement).
In 1958, the Supreme Soviet enacted new Fundamental Principles of Criminal Law and of Criminal Procedure, which greatly reduced the level of repressiveness; each of the union-republics then passed their own new criminal and criminal procedure codes. Essentially, the burden of proof on guilt shifted to the state; it somewhat approximated "beyond a reasonable doubt". The Principles also gave judges greater latitude/discretion to consider circumstances in sentencing. A confession was also no longer sufficient to establish guilt, now the establishment of guilt required other substantive evidence. Procurator General Vyshinsky's standard that probability of guilt was enough to warrant conviction was abolished too. A big change from this is the reduction in the maximum sentence; from 25 to 15 years, and the maximum term of exile from 10 to 5 years.
All that being said, the Principles did not formally recognize the presumption of innocence. Also, they did not resolve the issue of the restriction of the right to legal counsel during the preliminary investigation. The preliminary investigation bit is pretty unique and I'm going to give a very short description; basically, a suspect could be detained with no contact with an attorney (72 hours, could be held for an additional 7 days) while professional investigators prepare or drop charges. The process involved gathering evidence, questioning witnesses, and interrogating the suspect. If the investigators were convinced a crime was committed and that the suspect is guilty, they sought an indictment. What ended up happening was that 99% of those indicted were ultimately convicted. The Principles only gave the right to legal counsel during this phase to two categories of individuals; juveniles and the mentally/physically incapacitated.
Progress on the rights of the individual was a bit iffier, but there were still significant developments. The Statute of Procuratorial Supervision in the USSR of 1955 spelled out procuratorial powers of protest, representation, and established the rights of procurators to demand (from officials) information/documents concerning violations of laws. The 1961 Fundamental Principles of Civil Law and Fundamental Principles of Civil Procedure gave Soviet citizens more ability to initiate civil cases against state agencies. Judicial review was now permitted in some areas of interests to the average Ivan Ivanov, such as; imposition of administrative fines, disputes on state-owned housing, work-related grievances. Important areas of interest that were excluded were those regarding social programs around health care, pensions, education. Comrades' courts and informal tribunals were resuscitated to resolve minor disputes; a Model Statute of the Comrades' Courts in 1959 expanded their jurisdiction to include minor criminal cases that could be handled informally by social pressure (the social pressure aspect of the USSR is a whole other interesting aspect to consider). By the end of 1963, these relatively informal courts had handled more than 4 million cases per year. Their role started to decline again around 1967, when the professional legal establishment began to want to abolish these or incorporate them into regular judicial procedure.
All this started to change once Brezhnev came in (Krushchev was ousted in 1964). Brezhnev stressed legal regulation and marked an end to the aforementioned "utopianism"; the new Constitution of 1977 re-affirmed the central role of the party and the interests of the state.
For more reading and sources, please see:
Reforming the Russian Legal System by Gordon B. Smith (1996) - probably the best overview of the Soviet and post-Soviet legal history.
Justice in Moscow by George Feifer (1964) - this one is on the people's and comrades' courts, very interesting and funny.
The Judge in a Communist State by Otto Ulc (1972)
TDLR: It depended on who's in power. Human rights/political freedom, imo, hinge on relatively stable and fair rule of law.