Having the same problem with a Lumia 820.
Could it be a carrier, Digicel (in Trinidad) in this case problem?
I also have this problem. Digicel (Trinidad)
What if you do not have an older phone?Put sim card into an older phone types Nokia N85 and there you have the option to delete SMS from the SIM card.
Sent from my Lumia 1520 using Tapatalk
That just brings you to your message inbox. Does not delete anything on the sim card.Or open Storage Sense> email & Messaging > manage messages. You can delete individually or whole threads. That is the way it works on my AT&T 920, maybe you carrier has that option locked if it does not work.