The only thing negative when I applied for my mortgage was my student loan lates. I had some other baddies but they had either fallen off or been removed years before (2012).
As far as the student loans go, I believe I had 4 tradelines that all showed 120+ lates. Keep in mind they were pretty old at this point. My last late was September of 2011, and my mortgage application was approximately 5 years later. I did have to write a one sentence explanation for each tradeline explaining why I was late as part of my underwriting process. These lates are still on my credit report and will likely be there for the full seven years. Frankly, I don't think my score will go over 720 until they fall off.
Another thing to note is that FICO scores used for mortages are often not the same as ones given to you by credit card companies or even this website. You need to ask your lender what mortgage score they use. When I applied for my mortgage my FICO scores were all hovering around 710-722, but my mortgage scores were a bit lower at 680-690. It wasn't a huge deal, but it meant that I didn't qualify for the lowest conventional rate, which is why I ended up with an FHA. I really suggest checking out the mortgage board, it was really helpful for me when learning about the process.