The most renowned case surrounding sodomy laws is Lawrence v. Texas, but the road to that Supreme Court decision holds several other landmark cases. In 1976, the first suit to be brought to the Supreme Court regarding sodomy laws was Doe v. Commonwealth Attorney of Richmond. The Supreme Court upheld the state law due to a technicality. Several states throughout the 1980s challenged their sodomy laws in higher courts.
In many cases, however, the Supreme Court supported state sodomy laws. Notably, the Bowers v. Hardwick case in Georgia in 1986 where the court upheld that the sodomy laws in the state could be used to discriminate against homosexuals. This case became the justification for discrimination against sexual minorities outside of sexual acts. After that case, activists had a new strategy of challenging sodomy laws through the state courts saying that they went against the state constitutions. Since 1995, the ACLU has taken an active role in fighting against state sodomy laws. Unfortunately, even after the landmark Lawrence v. Texas case, the fight against state sodomy laws still continues.
Link: https://www.aclu.org/other/getting-rid-sodomy-laws-history-and-strategy-led-lawrence-decision