Runners in her hose?

Women's hosiery of that era were not yet pantyhose, but rather thigh-high hosiery held in place by garter belts. The nylon from which the hosiery was made was subject to having a defect occur, called a run, that would leave a visible line or streak down the length of the hose. Wearing such hosiery was at least mildly embarrassing and to some extent a sign of low social or financial standing since the woman was not able to afford better (or more) hosiery. The same thing can happen with pantyhose, but that was not what the Johnny Cash song had in mind.

1. a series of unravelled stitches, esp in stockings or tights; ladder

http://dictionary.reference.com/browse/run

A runner is just a "run", so called because the initial damage (e.g., a snag or tear) would "run" along the leg because the stocking is both knit and worn stretched over the leg, thus, worn under tension. When the the strength of the fibers was equal to the strength of the stretch the "run" stopped "running." But a small bit of damage could run up or down the leg extend and cause more visible damage.

Looked like this ...

As mentioned, this type of damage could occur in nylon stockings (= "hose"). The old image in the comment by Superbest seems to be defunct.

I don't know about the predecessor, silk stockings.