In the United States moored balloons (or kites) are governed under Part 101 of the FARs (a much neglected and very short corner of the regulations that also covers amateur rocketry and "unmanned free balloons" -- looking at your profile it's probably a section you'd find very interesting!).
In this particular case you're probably looking for Subpart B - Moored Balloons and Kites.
The FAA is mum on the specific engineering requirements of mooring systems - I expect they anticipate you will work with structural and mechanical engineers to design a tethering system that can safely hold the loads your moored object will place on it (or just tie it to the biggest, most structurally secure thing you can find, with a nice heavy rope and a proper knot).
There are a few things they ARE pretty specific about though.
- Some operating limitations (altitude, size, location, visibility).
- The fact that you have to tell them about it in advance
(At least if it's more than 150 feet AGL, per
- Lighting and marking requirements for the tether(s)
14CFR101.17, which basically says the mooring line must have pennants (daytime) and/or lights (nighttime) to ensure visibility)
- What happens to Balloons if the tether breaks
14CFR101.19, which basically says if the balloon breaks free of its moorings you need a "rapid deflation device" that gets it down in a hurry so it's not just wandering around up there).
Interestingly they don't seem to care so much if a kite escapes its moorings.