I also got affected by the interlacing problem. This is what happened:
My first iPhone 5S had a serial number that started with C39 and had a perfect screen, no interlacing and flickering at all (I knew where to look at since I had a iPhone 5 with interlaced screen before). After some weeks the glue on the display glass was bad, so the glass moved and I had to go for a repair. Since they can't repair any display defects on the iPhone 5S (thanks to TouchID) they offered me a replacement. The replacement I received had a serial number that started with DNP and had the worst display I have seen in the last few years. The lines were visible everywhere and all the time, which caused me headache and dizziness if I looked at the display for some minutes. I tried the test image and it flickered like hell. So I went back to the Apple Store for a new replacement. Luckily the Genius saw the lines as well and agreed instantly that this should not be. So he switched the iPhone again and my new model had a serial number that started with F2L. It had a slightly better screen than the DNP model but the lines were still visible to me and the test image still flickered. The genius offered me again to replace the iPhone as he still agreed that the display isn't as it should be. I insisted to get a replacement that starts with C again, since my first iPhone was the only one with a good screen. The genius then brought me a replacement with a serial that starts with C39 and tataaaa I finally got an iPhone with a perfect screen again (no visible interlacing lines and no flickering with the test image)!
I checked almost every iPhone 5S and also 5C in my local Apple Store (I tried like 20 phones and also 2 personal phones of 2 geniuses) and ALL of them had the interlaced lines and flickering! Also interesting was that all of these phones had serial numbers that started with F or D, there was no one with a C serial number.
The genius was really surprised that almost every screen got this issue and he agreed that there must be a serious quality issue with the new iPhone screens. He promised me he will report this internally to Apple/his bosses but couldn't tell me what/if Apple will do about this in the future.
So far I'm happy I got a good screen again and I was really happy with the genius since he took my problem with the iPhone seriously since the beginning and tried everything to find a solution. But on the other hand I'm really disappointed with Apple. I payed a lot for this phone (I bought it without contract) and for that price I expect a far better quality.