For US material the PF and PSE will be the most widely accepted. That can be further divided by type of material with the PF having the edge for classic US and more esoteric material. PSE grading services are still the most popular for, well, grading. Some houses only quantify PF and PSE certificates (Siegel comes to mind) Next would come PSAG certs. Kelleher in particular seems to utilize them a great deal for new work. Crowe, APS/APEX and Weiss are also out there.
For specific non-United States material a certificate from a recognized authority is likely the best bet. There are many ranging from Zagorsky for Russia to Greene for Canada. Research is called for. Look at past auction sales and see what certificates were presented.
I should add that while a certificate is usually not a prerequisite for being accepted for sale having one will increase the sales value exponentially in most cases and the purchaser at most sales will have the right to place the lot on extension so it will end up expertized anyway.
There are fake certificates but they are rare birds in my experience. The PF and PSE allow you to look up the certificate by number which pretty much precludes any shenanigans. I cannot speak to other certifying bodies for WW stamps but all newer certs have watermark features, embossed stamping, photos attached by adhesive, original signatures and so on.
For Terry, There are also mistaken certificates. I recently ran across a rather glaring error on an older PSE cert of a supposed US 443 pair which is a perf 10 coil. But the stamps (clearly shown on the cert, too, no mistaking the torn perfs) were perf 8.5. Making the coil pair in question a 412. The cert is real, but it is wrong.
Others have posted here occasionally about how expertising in past decades was not picking up faked coils and reperf jobs as reliably as recent expertising by the leading names seems to. So older certs are deprecated to an extent depending on what is being certified.
Signature stamps on the back of say French and other continental stamps have been know to be forged, but those are another matter.
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