Are search warrants public records? Does it matter whether an investigation ended up resulting in a prosecution? If so, does it matter whether the case goes to trial? If so, does it matter whether the prosecution is successful (defendant is convicted)?
If search warrants generally are public records, what is the best, most efficient, most compelling way to retrieve them?
Short answer: Maybe.
Long answer: The answer here varies from jurisdiction to jurisdiction.
Normally, the process goes like this: The application for the warrant is usually made under seal or otherwise in secret to prevent the target from trying to hide evidence. So before it's been executed, you can pretty much forget about accessing it. After the warrant is executed, though, there are differing answers to this question.
The Supreme Court gave us a test for this kind of question in Press-Enterprise II, which held that the First Amendment gives us a qualified right to access court proceedings and records. The right applies when public access makes sense using the "experience and logic" test: Has the Anglo-Saxon experience typically been to provide access, and does logic tell us that access has beneficial effects for the judicial process?
But lower courts have disagreed about how to apply the test. The Eighth Circuit allowed access to warrants in a defense-industry corruption investigation in In re Search Warrant for Secretarial Area-Gunn, but the Ninth Circuit denied access in to warrants in the same investigation in Times Mirror Co. v. U.S. I think, though, that the Ninth Circuit decision would have come out differently if the investigation had already ended.
Just to mix things up further, the Fourth Circuit has also allowed access, but based on common law principles of access, rather than the First Amendment. That was Baltimore Sun Co. v. Goetz. Same in the Second Circuit: In re Application of Newsday, Inc.
Individual states also have their own rules, but those are of course subject to limits under the First Amendment.
When I wanted a copy of a warrant, I would go first to the clerk of the court whose judge signed the warrant. I would tell them what I was looking for, and I pretty much always got it.
I would rarely submit a FOIA request, especially if the warrant was issued by a federal agency. Those requests sit in a queue for months or years without being reviewed, and the agency virtually always denies the request anyway. When law enforcement agencies and courts have copies of the same record, you're almost always going to have better luck getting access from the courts, which are set to open by default. If the court denies the request, try again after there's an indictment, and again after the trial.
In the United States, the right to a public trial means that the prosecution can have no surprises in the evidence that is presented to the court. The Defense is entitled to see every shred of evidence (including evidence that is harmful to the Prosecution's Case) and is given access to witnesses to question their account of the events. Depending on the nature of the warrant, the person it is executed against will see it at the time of execution... though some warrants, like wire-taps, may not be presented right away (and due to the Top Secret nature, FISA warrants may not be seen at all).
If you are not party to the warrent, the best way to view it is a Freedom of Information Act (FOIA) request made to the agency of government that served the warrant but there could be some restrictions on the release of warrant, especially if the case is being followed by the press.
