It really depends on what area you're talking about, but for the sake of answering this easily we'll discuss European popularity. Cotton had been brought in the early 17th century fabric trades from Arabia and Asia, their particular cotton plant having a loose, not sticky seed and being therefore easier to harvest. It gained massive popularity in the 18th century in Europe, England even going so far as to ban 100% cotton being imported from 1720-59 to prevent damage to the wool trade. Though production of cotton continued, wood-block printing began about 1676. Printed cottons, called Chintzes, or Indiennes in France, were hugely popular as well as more particular textiles like Nankeen, an unbleached, yellow, cotton originally from Nankin China. Others included mull, velvet, pique, percale, organdy, muslin, and mixes like seersucker, Virginia cloth, and jean.

Despite it's popularity, it certainly didn't oust another fiber at this point; linen, silk, and wool continued to be popular. However, things do change when the cotton industry comes to America. Prior to the cotton gin, the type of cotton that could be grown in this area was very difficult to harvest. It has a sticky seed that required a person to individually pick it out. Very time consuming and tedious, keeping it a smaller scale industry. With the cotton gin invented in 1793, the seeds were shaken out by a machine and the processing sped up rapidly. Cotton takes off in production and popularity.

If cotton replaces any fiber, it might be linen (flax). Linen made up the majority of undergarments and light weight outer garments before cotton (wool made up some as well). Today, the processing of flax fibers to turn them into threads for weaving actually occurs on machinery made for cotton fibers. It really doesn't work, since the flax fibers are 3-4 feet long and cotton is 3-4 inches. So, they chop up the flax, essentially destroying it's strength and leaving the consumer with an easily wrinkled fabric as well. Cotton is also built as a plant to eventually rot away so the seed can be set free. Flax comes from the vascular system of a plant, so it's meant to wick water and specifically not rot. Meaning, sweat on the body gets wicked away and evaporated. All of this doesn't seem to deter from cotton's popularity, however.