If you own an iPhone 6s or 6s Plus, or will be getting an iPhone 7 or 7 Plus, iOS 10 lets you 3D Touch some lock screen notifications to respond to a message without even unlocking your phone. This is definitely a handy feature, and it should save a lot of time in some cases, but there are some obvious security concerns.
So if you're worried about someone being able to respond to your incoming emails or texts without having to unlock your phone, head to the Touch ID & Passcode menu in Settings. From here, enter your passcode, then scroll down and disable the "Reply with Message" option. From now on, nobody will be able to respond to notifications on your lock screen.
While you're at it, you may want to take things a step further. The new iOS 10 lock screen notifications can do a lot more than they used to, so it would be a good idea to limit the apps that can show notifications here at all. If you'd like to prevent certain apps from being able to show lock screen notifications, just head to the Notifications menu in Settings, select any app, then disable the "Show on Lock Screen" option.
Keeping with the subject of the new lock screen giving away too much information, you may also want to disable iOS 10's lock screen widgets. It's a fairly simple process, and I've outlined it at the following link.