I tried a lot of these solutions - (iPhone 6s / Verizon)
1. hard reset
2. airplane mode, off and on
3. reset network settings
4. popping the sim card out
If you go to settings, about: Look at what the carrier says; in my case Verizon. Tap on the carrier and the next item show PRL. If PRL shows "1", then you are not on the correct "Priority Roaming List", and that is what is causing the phone to not be able to connect to the carrier. I think it is the same for the other carriers also.
Here is what worked for me step by step:
1. Turn your phone off.
2. Take out your sim card, leave it our and power phone back on.
3. Go to settings, General, reset, and Reset network settings.
4. Once the phone cycles back on. Turn your phone off by holding the power button.
5. Once powered down completely, re-insert the sim card.
6. Power back on.
7. Your phone is connected back to the network.
Hope it works for you!