As the comments have mentioned it depends on components, size of board, etc.
As for the board itself, if the standoffs are that irregular then your own strength is going to let you down well before you reach the board's breaking point. A quick look at the datasheet for the boards I get fabricated (from pcbtrain.co.uk, see link below) shows the warp strength is in the region of 50 Kpsi (400+ MPa).
Surface mount components are going to give you the most trouble with bending boards, as others have said. But don't forget things like large electrolytics or bulky connectors. A flexed board is going to have a weak spot. And you'll end up with an oversized capacitor coincidentally placed right on that weak spot. And that oversized cap will sit very close to the chassis that's housing your lovely, new (flexed, but who cares, right?) PCB. And you're going to drop it. And, like a piece of electronic toast, it's going to land upside down, right where that capacitor is pressing against the board. And the board is going to say...
Nothing, because it's broken. At my previous job I have seen it happen regularly in certain mass-produced small electronic devices like pocket and portable radios; the break is ALWAYS in the same place for a given product, and it's invariably a catastrophic failure. In some cases, if the flex is that bad on the board (which you really only get if it's screwed into standoffs, not clipped) you're actually better off not using the anchor point that causes the greatest anomaly.
Datasheet ref: http://www.pcbtrain.co.uk/cp/uploaded/VT481TDSrevA18_9_13.pdf
The Velcro is probably no worse than any other plastic (nylon) that is close to your circuit, as far as ESD is concerned. It may be a problem if you frequently rip the Velcro off, as you may cause the friction to generate a charge.
But a big problem with Velcro is that it does not last very long. The self-adhesive side will stick OK to begin with, but will dry out and let go in a comparatively short time. I've seen this happen on a regular basis with electronic modules fixed to housings with Velcro. They all came back with the modules dropped off but the Velcro-to-Velcro interface intact. The Velcro had just come unglued from the inside of the housing.
I'd probably go for M3 or M4 stainless steel socket head cap screws. They're strong, corrosion resistant, and readily available around the world. If you google DIN 912 A4, (DIN 912 meaning a metric socket head cap screw and A4 meaning stainless steel), you'll find lots of suppliers.
One such supplier: http://www.mcmaster.com/#din-912-a4-socket-head-cap-screws/
I'd have those screws pass through nylon unthreaded standoffs and thread into whatever you're mounting the board to. Here's what I'm talking about: http://www.mcmaster.com/#catalog/116/3240/
You might also try snap-on plastic standoffs like these: http://www.mcmaster.com/#catalog/116/3245/
They're designed to snap into a 5/32" (3.97 mm) hole, so they should just fit.