There is, not surprisingly, debate on the Court as to what sort of places and fora should be considered non-public.  Greer v Spock considers restrictions on handbill distribution and speechmaking at Fort Dix Army Base in New Jersey.   Dr. Benjamin Spock and others challenged restrictions on speech, arguing that since the base welcomed civilian speakers and had entertained musical concerts and other expressive activities in the past, it should be considered a public forum.  The Court, however, disagreed.  Fort Dix was a non-public forum, so restrictions on speech need only be reasonable and viewpoint neutral in order to pass First Amendment muster.  In Greer, the Court (over dissents) concluded that the challenged restrictions on speech served the army's goal of keeping its troops focused on military preparedness. 

Cornelius v NAACP Legal Defense and Education Fund also found the Court divided over whether the forum in question should be considered non-public.  In Cornelius, the forum was not a place, but a federal fund drive.  Each of over 200 charities accepted for inclusion in the federal employee fund drive was invited to prepare a 30-word description of their charitable activities to encourage payroll donations that would be directed their way.  Several excluded charities sued, arguing that the critieria used to exclude them from the fund drive violated the First Amendment.  They also attempted to argue that the fund drive was a designated public forum.  The Court, however rejected their argument, finding the drive to be a non-public forum. So long as the criteria used for determining eligibility for participation in the fund drive were reasonable and viewpoint neutral, the Court said, the First Amendment was not offended.  (The Court, finding some "evidence" of viewpoint discrimination, remanded the case to the trial court on that issue.)

In Board of Airport Commissioners v Jews for Jesus, we see that even in what is arguably a non-public forum (the Court says it need not decide whether Los Angeles International Airport is a traditional public forum or a non-public forum), speech restrictions may run afoul of the First Amendment.  The Court finds LAX's ban on "all First Amendment activities" to be so overbroad as to offend even the lax reasonableness test applicable to the non-public forum.

A co-founder of the Minnesota Voters Alliance displays a T-shirt deemed unacceptable for wearing while voting.

In the 2018 case of Minnesota Voters Alliance v Mansky, the Court considered a Minnesota ban on the wearing of certain types of campaign-related apparel in the immediate vicinity of polling places.  The ban reached everything from buttons and T-shirts promoting specific candidates to clothing items that were closely identified with campaign issues (e.g, an NRA T-shirt or one that included the text of the 2nd Amendment, Tea Party T-shirts, etc).  In his opinion for the Court, Justice Roberts noted that a polling station is a government place established for a specific purpose (voting) and thus is a non-public forum.  Therefore, the Minnesota law should be upheld if it is reasonably designed to further a significant state interest such as keeping peace among voters, providing a dignified atmosphere for voting, avoiding undue pressure on voters, etc.  While the Court found these interests to be significant, it nonetheless struck down the Minnesota law, finding its application to be too arbitrary and giving too much discretion to election officials to decide what clothing was too political to be permitted.