Q: constantly showing no service

Wow, so big a question, so little information. Did you purchase the phone new from a carrier or Apple? Or was it used? Or did you buy it from a 3rd party who SAID it was new? Did it ever work? Is it locked to a carrier? If it is locked, are you using it with the carrier it is locked to? If it is unlocked did you purchase it from Apple directly, and does the receipt say it is unlocked? Likewise for a purchase from a carrier. Have you established an account with the carrier? Have you called the carrier to verify that they account is active and provisioned correctly for an iPhone?

A carrier is the company that provides your cellular service. You must have a contract with a carrier to get service. You must pay a monthy fee for this service, or have a prepaid SIM card that has remaining balance on it in your phone. When you had the phone replaced did you take the SIM card out of the old phone and put it in the new one? If you did not you must contact the carrier (cellular company that provides your service) and have them register the new SIM card on your account.

I am on O2 and for the last 3 weeks since I had my screen replaced on my 4s, I am losing signal and often find my phone has "No Service" when there should be a full signal from my carrier. I have followed all of the above advice and apple's instructions but still this is happening making my phone unusable at home!

How can I fix this problem?

Whoever replaced your screen did not reconnect everything properly. Apple doesn't replace screens, they replace the whole phone, so I would take it back to the place that replaced it and complain.