Hi,
I had a low-lying placenta at my 21 week scan. I was also told that in the overwhelming majority of cases, the placenta moves away from the cervix as the uterus expands, and that I really shouldn't worry about it.
I'm now 35 + 3 and will go for my scan to check it's moved on Monday. I've been told to expect that everything will be fine for a normal delivery, but that if it hasn't moved, the worst I should expect will be that I'll have to have a caesarean.
I've also been told that there is a world of difference between a low-lying placenta and placenta praevia. The latter is where the placenta is wholly or partly covering the cervix and can be more prone to causing bleeds during pregnancy (due to pressure from the baby) and if the placenta hasn't moved by 36 weeks or thereabouts, it would certainly mean a c-section.
However, low-lying can just mean the placenta is a bit close to the cervix at 21 weeks for comfort - in my case, it was 1.5cm, with the hospital preferring a 2cm gap for vaginal birth. So it's worth checking just how 'low-lying' your placenta is.
The main issue is to ensure you report any bleeding immediately to the hospital - they'll want you to come in straight away to check it's nothing serious. I had some bleeding around 26 weeks - turned out to be totally unrelated.
When I first found out about this, I worried terribly, but the reality is that it's very common and will almost certainly resolve itself.
Good luck with the rest of your pregnancy, and try not to fret. I'll try to remember to post on Monday with the results of my scan - here's hoping I get the all-clear!