VODAFONE (LSE:VOD) - is the biggest mobile network provider in the world, even being co-owners of VERIZON Wireless in the USA. They’re usually one of the first networks to pop-up when you delve into the nerdy technical settings of your mobile phone (Cellphone) to scan for available networks.
O2 (company history) is the other one mostly likely to pop-up first when you scan for available networks. Were first in the UK allowed to offer the iPhone.
ORANGE (company history) - all I can tell you is that they exist, sound like they have some pretty good deals. Unfortunately I’ve also heard several stories of people who have to go for a bit of a long wander to try and get a signal through this network.