Is your iPhone set to use the Full Preview for Screenshots? This is at Settings > General > Screen Capture > Full Screen Previews.
With this on, you will be presented with the Preview to give you a chance to make changes and then you will need to tap the checkmark on the top right of the screen to choose where you want to save it. If you exit out or cancel by tapping the "X" in the upper left, then nothing is saved at all.
With the Full Screen Previews turned off, then it works similarly to the way it always has where you will just see the thumbnail in the lower left corner for a couple of seconds, then it will automatically save to your Photo Library. Just make sure you do not tap that thumbnail, otherwise it will open the Preview and you will have to choose where you want to save it.
Assuming you have Full Screen Previews turned off and did not tap the thumbnail after taking the screenshot, then it is in your Photo Library after the thumbnail goes away. Photos can be viewed either with the Library tab on top or Collections. When using the Library you can see what is included by tapping the Filter/Sort Button (3 lines) on top of the Library Screen, then choose View Options. You should have a check next to Screenshots there if you want to include those in your Library view, otherwise you can view them in your Collections tab either under Recently Saved or Media Types.