Most appearances in World Records

[Powell]

Koch has set more than 20 WR's (I'm sure of it!)

If you only count official WRs, I have her at 16 (including relays and equalled records). She definitely has over 20 if you include indoor records, which weren't officially recognized by the IAAF until 1987.

[gh]

As to total WRs, then DJ is correct (though it's close:
Nurmi - 22 official + 13 unofficial = 35, plus indoors 5 unofficial = 40

If you want to get into the weird world of things that used to be considered WRs, note that Nurmi set indoor "WRs" at 12 different distances during his U.S. indoor tour of 1925:

Mile, 1.25M, 1.5M, 1.75M, 1.875M, 2M, 2.25M; 1500m, 2000m, 3000m, 4000m, 5000m!

That was a 4-month tour in which he ran approximately 68 races in the U.S. In his first 12 days in the country he set WRs at 7 distances in 5 different races. Different times.