Android has great notifications, but you can always make them a little more efficient with some customisation.
SlideLock sucks in all the notifications from your device and puts them on the lockscreen so you can triage everything seeking your attention without unlocking the device. This app also has a few neat UI tricks that make it one of the more competent lockscreen replacements on Android.
SlideLock works best on Android 4.4 KitKat, but should be usable on any device running 4.1 or later. Because Android doesn't have a specific API for replacing the lockscreen, you'll need to disable the built-in option and activate SlideLock separately.
The app will pop up each time you wake the device, but it can be bypassed by pressing the home button. Android 4.4 allows the on-screen buttons to be hidden (if your device has them) to make it feel more like a real lockscreen.
The device can be unlock with a swipe, but each notification can be swiped to dismiss or open each one. You can also configure which apps produce lockscreen items, and how much information is displayed. The app plugs into the accessibility service or notification listener (Android 4.3 and higher) to replicate all your notifications on the lockscreen. If you're using the notification listener it can also dismiss notifications from the shade when you clear them from the lockscreen.
The basic functionality is supported in the free version, but a small in-app upgrade is needed to unlock all the features including a custom status bar, custom wallpaper, and more.