That really depends. Carriers make changes to the OS but as a general rule, it also depends on the OS you were running.
The first thing: did you have an external SD card in it? If you did, remove that and check to see if your screenshots are there. More than likely, they won't be there because screenshots (using the OS feature rather than an app) usually are stored on the device memory rather than an external card, but some may save them to the external/removable/expansion card.
If they are on there, you are in luck. Just pop the card into a computer to check. See if they're in a folder called Screenshots or Screencaps, or if they're in a subfolder of the DCIM folder (the place where pictures taken with your camera are stored).
If they're on the device memory and you didn't ever make backups, you could try connecting your phone to a PC using the usb cable, but if you can't access the phone from there or browse the phone's memory, you may have to see if a tech professional can recover them, unless you know how to use ADB Bridge and a program like Eclipse or something.
My S5 saves screen captures to the device memory, but my old phone (StraightTalk Galaxy Proclaim) saved them to the SD card. But that's the first place to look.
If you have a Windows machine, you can search for file types, so you can actually search that card for image files only, but I forget the search syntax.