You know, the small details like this are more and more important now that "fake news" is a more serious and dangerous phenomenon. It's important to realize what ACTUALLY is involved in a federal court case, or this or that executive order, or a confirmation hearing. Thanks, Chanku.
@PB: Oh, no problem. Nice to know someone liked my post. Personally I just prefer to try and have people understand that while they may disagree with it, there is a point to the actions of a court and there are certain things a court can or can not hear or do. It is important to have some grasp on that.
The only way the supreme court would ever heard a case involving a suspension of an executive order is if a suit made its way to the Supreme Court, and that suit was over if the court has the jurisdiction to do so. This is highly unlikely as the court would more than likely rule that the suspension is legal.
In fact the ability to issue an emergency injunction and to get a stay against something like this is important. (Note technically the 'suspension' is actually an injunction against the order's enforcement. Essentially it directs everyone who is to enforce it to not enforce either whole or a part of the order, dependent upon the request granted. This is because an injunction is against a person, while a stay is against a court.) The ability for these to be enforced is important as it allows the court to prevent harm due to a law or action until a suit's completion. For example let's say I get a law passed that says that Person A is forced to give up everything they own. Well they could sue and request an injunction against the enforcement orded and prevent its enforcement until the lawsuit is invariably completed, more than likely against the law. A stay is important because, let's say that a ruling says the law is illegal and it is appealed, the court can stay the ruling, which means that the law can not be struck down until the completion of the case (at least if I understand it, I may be incorrect here. So do not quote me on this). These measures exist to prevent, or migitate, harm caused to one of the parties. Keep in mind a court can do both stay a ruling and issue an injunction against the enforcement of something. Also emergency injunctions are typically used when there is an issue that can cause gross harm to one of the parties, generally that harm may be irrepariable. In this case, there was an injunction issued preventing the removal of people from the US pending the outcome, as returning anyone may cause irreparable or gross harm upon that person. If the order is upheld in the end, these people could be made to leave or be barred from Re-entrance. When a case comes before a judge, especially concerning the legality of laws, they aim to minimize harm suffered by the parties, and if there is potential for one party to face gross/irreparable harm then the court will usually act to prevent that. Now keep in mind i am not up to date with the court proceedings and the like, so this information may be incorrect or outdated.
Also order of stay or order of injunction is also generally public, so you can read the order related to any of these cases. I do suggest you read them so you can understand the Judge's rationale, as they usually contain the judge's reasoning for granting/issuing an order. Also please keep in mind thay I am not a lawyer nor am I your lawyer, or anyone's lawyer. I am not licensed to practice law in any country or subdivision thereof. Do not take my statements here as legal advice, but instead information to learn from. Since i am not a legal professional, I may have made mistakes and if I have please point them out, so I may correct them and migitate the spread of false and incorrect information.
EDIT: I also wish to add something. If the case would be dismissed due to improper filing or something similar, then it would more than likely be dismissed without prejudice, allowing the case to be refiled in a lower court. A ruling of this would vacate the rulings of lower courts on the case, unless it is refiled. Now a case can also be dismissed with prejudice, which means that the case can't be refiled or heard, and this would also technically vacate the lower rulings. The vacation would occur because the lower courts shouldn't have heard it for one reason or another. Also the court could rule that they lack jurisdiction, which is VERY unlikely, but could also act as a vacation of lower rulings, due to the lower courts more than likely lacking the jurisdiction as well. Also keep in mind I am assuming that the Supreme Court acts similarly to the lower courts, and has these powers. They may very well not, and in which case I am essentially talking out of mt ass and please disregard this. If this is incorrect or flawed please let me know so I can fix it.
EDIT 2: Some fixes regarding wording and grammar. This is what I get for posting from my phone. Also keep in mind that the courts try to minimize harm to all parties, however gross/irreparable harm takes precedent. Also stays can be issued against the court that issued the stay. For example a court could stay a trial until the completion of another trial. This may be because the matters are related, and one outcome could affect the other, or render the suit moot in the first place. Again this information may be incorrect or outdated. Please let me know if it is and how it may be corrected so I may do so.