I'm not good at navigating some parts of the NGS website ( https://geodesy.noaa.gov/ ). I'm looking for some sort of manual or specification that explains how they go about establishing horizontal physical marks. Ideally something written in the 21st century.
You may be overthinking this. NGS doesn't establish passive marks in the 21st century, so I don't think that you are going to find a lot of recent writings.
The datasheets will note that marks have been established by "GPS Observation", "Classical Geodetic Methods", or "scaled from a map". For those established by GPS Observation keep in mind that perhaps no marks have been established after the HARN/HPGN programs of the mid to late '90s - necessarily using the technology and constellations available at the time. That means GPS only, no GNSS. For those established by classical geodetic methods the only thing you really need to know is that their positions are going to be accurate to ±1 foot relative to current OPUS results - at best. The whole story of the triangulation program is interesting history, but it is probably not that necessary to know the details of for your day to day work.
Nevertheless, check out the 1984 standards for such classical work at https://www.ngs.noaa.gov/FGCS/tech_pub/1984-stds-specs-geodetic-control-networks.pdf
Norman gave some great advice. Also some passive marks have been re-observed though Bench Marks and HARN and many others since GPS. All of this will be acknowledged in a data-sheet for marks. If you are wanting to set new passive marks there are guidelines if you wish to bluebook them. If you are not but wanting to achieve equal accuracy to NSRS the book pdf norman said is a good one but also look at the new FGCD guidelines for standards but it will not go into HOW so much as the old manual but what needs to be done to achieve a certain standard etc. Also for GNSS only GPS so far so . A great class is OPUS Project Management they usually touch on NSRS and best practices for better relative positioning as well. Thats free online webinar. Sign up for the next available and you will gain some good knowledge as well.
Are you looking for the standards for entering a mark into the NGS Integrated Database (IDB)?
You might be looking for this document:
https://www.ngs.noaa.gov/INFO/Policy/files/062012_Data_Submission_Policy.pdf
It has links to other standards and specifications documents, but is basically the current guidance for adding a new mark (and observations) into the database.
Maybe the OP is asking about the physical construction of geodetic control marks. If that's the case, he should check out "Bottles, Pots & Pans? - Marking the Surveys of the U.S. Coast & Geodetic Survey and NOAA" by George Leigh, NOAA Corps (Ret'd). It has a lot of information about physical marks, and lots of references to more.
https://www.ngs.noaa.gov/web/about_ngs/history/Survey_Mark_Art.pdf
Thanks for the responses. The links are interesting. It also suggests that I haven't missed a more up-to-date version of any of these.
There's also this one on setting horizontal/vertical marks for PACS and SACS, which was definitely written in the 21st century:
https://www.faa.gov/documentLibrary/media/Advisory_Circular/150-5300-16B.pdf
See NGS 2010 Bench Mark Reset Procedures -- https://geodesy.noaa.gov/PUBS_LIB/Benchmark_4_1_2011.pdf
Pages 12-17
This might be a little relevant, because I volunteer with the Civil Air Patrol, and there are vertical control benchmarks in an area we can walk to from our hangar, without being visible to pilots. (Not "hiding", it's just a matter of safety. They have enough to think about without seeing a person walking within the security fence.) This could be useful for demonstrating the lack of accuracy of recreation grade handheld units.
