What is the time frame, or rule of thumb for replacing rigging used in fresh water?

What about the attachment points on the mast and hull.

Some sailors replace their rigging every couple years. Usually it's the racing crowd that sail in saltwater who are mostly concerned about a dismasting from a broken shroud. I had a 1976 H16 (up to 2 years ago) with original rigging and didn't have any trouble with it. I did replace the 22 year old rigging on my Prindle 18 this year, but I'm not sure it was necessary. If the rigging looks good (no corosion at the joints, no frayed cable) it's somewhat of a roll of the dice. Sometimes, defects don't show up at all externally.
Jack

For shrouds: every 5 years is a reasonable schedule, more often for racers.

I replaced a mast tang once on an old mast (20+ years). The tang steel had obviously deformed (at the shroud/forestay shackle) and had begun to crack. I have also had the shroud anchor bar (piece under the hull lip) crack. I believe it was the result of salt water corrosion. Don't know if it is a common problem but it is worth checking once a year and also check the fiberglass lip which can wear and result in a top deck failure.

Like Jack I had a old boat with original lines for a long time. It all depends on use, storage, water. Freshwater is a good thing. In saltwater I have seen the anchor bars go south. The mag. schedule for replacement is every five yrs, but then again they are the ones getting your money.

I just noticed the other day that one of my shrouds has a sort of spiraled "rib" showing through the plastic jacket. Does this mean that one of the strands of the cable has snapped?

Does not sound good. Does it run the length of the wire? Take a good look at the two ends. How old is the wire?

I think it comes down to if your racing or not because if your racing you really do want stiff wires but if your out there just for some fun i would say replace them after 7 or so years but for racing nothing after 5 years. I just bought a new hobie and ill be changing them every 4 years because i want a good and fast boat. It comes down to personal preference.

I find it's always best to replace them just before they break.