PCB Design Verification
The desire is for the design to match the PCB layout and the PCB layout to match the schematic.
The most important thing is to work with the assembly house on ways/practices to avoid errors. The way I do this is I have a mechanical file with all component values and a part number, that way the information can be overlayed on the gerbers and it's easy to check where the physical component goes. The other thing is designators, and making sure they are in readable locations on the board and that the designators in the BOM match the PCB. I also have a note that the 'assembly house' reads in all my files and have been given special instructions to read each note in the file before processing/assembling the board.
I also have a guarantee from the assembly house that they fix all errors if they haven't assembled the boards right, that gives them extra incentive (but adds cost)
The biggest problems with assembly houses that I have experienced are:

Backwards components and not following SMT pin indicators

Solder joints between parts

Leaving out specially marked items on BOM or installing them incorrectly

A PCB with a plated through hole that was not tested (and should have been)


Most of the time they get it right, and usually they contact me with assembly questions before the build.
If you're really concerned about a build, make sure you do a prototype run of a few boards and test them before manufacturing 10-1000's of boards. I prototype my boards in house, normally I can turn a board around in 1 day after a PCB manufacturing time of 4-6 days. Sometimes I send out the prototype run for assembly and get 5 done, then release them to manufacturing for order.
I don't really concern myself with all of the industry specifications, because even with all of the ISO and IPC standards the assembly house still makes mistakes, and I don't want to spend 10% of my time reading, I'd rather be designing and fix the mistakes when they show up.
For higher volume designs, every detail matters as even small differences can affect the yeilds of correctly working designs. Things like soldermask size and pad size being off by a few mils can cause a small percentage of the parts to fail in the assembly process (think tombstoning and simmilar SMT failures). Most board houses/assembly houses are more than happy to help with issues or preempt issues through correct design. In the high volume cases work with your fabricators.