One of President-Elect Donald Trump's priorities when appointing a Supreme Court Justice to replace Justice Scalia is to select a justice who will "defend the Second Amendment." Commentators forecast that Trump's nominee will be pro-Second Amendment, and Trump's post-election statements suggest that the Second Amendment will be a key factor he considers in appointing somebody to the Supreme Court.
Those who anticipate big changes in Second Amendment law may need to curb their enthusiasm for the moment. In order to expand Second Amendment protections in any significant way, the Court must first agree to hear a Second Amendment case. This is something the Supreme Court has been loathe to do, with Justice Scalia and Justice Thomas occasionally protesting the Court's refusal to take on cases involving gun restrictions.
If Trump manages to appoint somebody who is essentially identical to Justice Scalia, this will only leave Trump with two justices who have been willing to dissent from denials of certiorari in Second Amendment cases -- Justice Thomas and whoever replaces Justice Scalia. Perhaps Justice Alito can be added to this group based on his ideology and his recent opinion concurring in judgment in Caetano v. Massachusetts (although there is a difference between the decision to grant certiorari and the decision to write a separate decision concurring in judgment).
Even with three justices willing to grant certiorari in Second Amendment cases, Trump's Supreme Court would still fall short of the four justices necessary to get the Court to hear Second Amendment cases in the first place. For this reason, unless Trump has the opportunity to appoint an additional justice, it is unlikely that there will be any immediate, significant changes to Second Amendment law.
Lastly, to add a bit of political prediction nonsense to this post -- if defending the Second Amendment is indeed a priority for Trump and if he values taking a more middle-of-the road approach when appointing a Supreme Court Justice, I would bet that Judge Diane Sykes (currently on the Seventh Circuit Court of Appeals) would be Trump's choice. The most recent commentary I've seen (and predictions of others who probably pay more attention to this than me) suggests that Judge Sykes or Judge William Pryor (of the Eleventh Circuit) are the most likely nominees. Judge Sykes may be a less controversial choice than Judge Pryor and Judge Sykes has a positive track record when it comes to the Second Amendment. Those wondering about her stance on the issue should look no further than her opinion in United States v. Skoien, and her dissent to the en banc Seventh Circuit opinion in Skoien that overturned her earlier opinion.
The Second Amendment will be one of many issues to consider as Trump nominates a Supreme Court Justice. But however Trump's initial nomination turns out, I do not expect that the Supreme Court will take any cases that lead to any significant Second Amendment shifts. This will most likely be something to watch for if Trump has the chance to nominate a second justice.