TexasFilly wrote:
Somebody asked me a good question last night and I do not know the answer (with certainty), as I have never filed an Application for Emergency Stay with SCOTUS (I'll bet not many lawyers have):
Does a Supreme Court Justice actually physically sign a stay (or denial), or Is their name printed (typed)? Anyone?
Well, I do have some experience with this (appellate work, including to SCOTUS, is 90% of my practice and, I'm sorry to say that I'm very well acquainted with the denial of stay applications

). And the answer is "Yes," the Justice actually signs the application for stay--at least in the case where s/he denies it.
SCOTUS rule 22.4 applies here:
Rules of the Court wrote:
Rule 22. Applications to Individual Justices
1. An application addressed to an individual Justice shall be filed with the Clerk, who will transmit it promptly to the Justice concerned if an individual Justice has authority to grant the sought relief.
2. The original and two copies of any application addressed to an individual Justice shall be prepared as required by Rule 33.2, and shall be accompanied by proof of service as required by Rule 29.
3. An application shall be addressed to the Justice allotted to the Circuit from which the case arises. An application arising from the United States Court of Appeals for the Armed Forces shall be addressed to the Chief Justice. When the Circuit Justice is unavailable for any reason, the application addressed to that Justice will be distributed to the Justice then available who is next junior to the Circuit Justice; the turn of the Chief Justice follows that of the most junior Justice.
4. [highlight]A Justice denying an application will note the denial thereon.[/highlight] Thereafter, unless action thereon is restricted by law to the Circuit Justice or is untimely under Rule 30.2, the party making an application, except in the case of an application for an extension of time, may renew it to any other Justice, subject to the provisions of this Rule. Except when the denial is without prejudice, a renewed application is not favored. Renewed application is made by a letter to the Clerk, designating the Justice to whom the application is to be directed, and accompanied by 10 copies of the original application and proof of service as required by Rule 29.
5. A Justice to whom an application for a stay or for bail is submitted may refer it to the Court for determination.
6. The Clerk will advise all parties concerned, by appropriately speedy means, of the disposition made of an application.
What we don't know (because that information in not necessarily included on the docket notation, although it's not terribly unusual for applications denied "without prejudice for renewal" to show that notation in the docket...if there is no "with/without prejudice" notation on the docket with the application denial, you just don't know unless you're a party/attorney in that case) is whether or not Justice Thomas denied Oily's application for stay
with prejudice (again, see Rule 22.4 for why this matters). Oily would know (it is part of the notification of disposition she would receive from the Clerk), but I seriously doubt that she even knows what "with prejudice" means, or that she would care, if someone explained it to her.