I would like to add to this. There have been several posts about Reminders not working with IOS 10, and while many of the questions show as "solved", it's clear that the issues have not been solved. Since updating my iphone 6 to IOS 10, and then to 10.0.2, reminders will not show up on my lockscreen. I too have now missed several important reminders. The solved answers direct users to toggle the options for interacting with the lockscreen. Here is one of the answers that has been posted:
In iOS 10, you can interact with notifications on the Lock screen to catch up on news, respond to messages, and more without unlocking your device.
Here are some ways you can view and interact with notifications from the Lock screen:
To view your lock screen, press the Home button or Sleep/Wake button. If you have an iPhone 6s or later, lift your device to see the Lock screen.
To respond to a notification, swipe left over it, then tap respond. If you have an iPhone 6s or later, use can use 3D Touch. Just press deeply on the notification, then respond.
To open the app to respond, swipe right over the notification.
To be clear, this does not fix the issue. Previously I was able to set up Notifications for reminders in the "notifications" portion of Settings. Now it doesn't even show up there (in other words, the reminders app is not listed as an app in Notifications). Reminders now do not alert me AT ALL. There is no alarm, no notice, no pop up screen on the lockscreen, no notification in the form of a banner while i'm using the phone. Nothing. I have to literally go to the Reminders app either on the lockscreen, or open the app, to see that i've missed my appointments. The issue is both with scheduled reminders (time/day) and location reminders.