SCOTUS hears the final 13 cases of the term during the last two weeks in April. Gorsuch will not be able to vote on cases that were argued before he arrived at the court, but he may have a decisive role to play in an important freedom of religion case to be heard April 19.
For over a decade, the Court has been divided between four more conservative justices, four more liberal justices, and Justice Kennedy, the swing vote on nearly every issue. Justice Scalia's sudden death nearly 14 months ago obviously changed that balance: over the last year, the best outcome the conservative justices could hope for in most cases was a 4-4 tie, if they could attract Justice Kennedy's vote. So Gorsuch’s confirmation restores the state of affairs that existed on the Court before Justice Scalia's death –- which means the answers to a lot of pressing legal questions are again in Justice Kennedy's hands.
The most anticipated case in the April sitting is probably Trinity Lutheran Church v. Comer, a case about whether a state constitutional provision that prevents state funds from going to religious institutions violates the federal constitution –- both the clause protecting the free exercise of religion, and the clause guaranteeing the equal protection of the laws. Here a church that contains a playground applied for a state program that helps non-profits re-surface their playgrounds. The church was denied access to the program because of its status as a church, and it argues that this is unconstitutional.
Other big cases to watch right now are the various challenges to the president's second travel ban executive order. Both the Fourth Circuit and the Ninth Circuit will hear arguments in May on the constitutionality of the travel ban. Whatever happens in those cases, the losing party is virtually certain to seek Supreme Court review. Although the court doesn’t typically hear cases between April and October, it's certainly not unheard of for it to do so -- and I think it's quite possible here, in particular if the administration loses and asks the court to act quickly. The court could also rule without hearing arguments.