The very purpose of the SIM card used to be to store contacts, since that was the only place you could store anything on cellphones. Now that the data can be stored internally, or to Drive (on an Android phone, which is the best place to store contacts), contacts shouldn't be stored on the SIM card (because if the card goes bad, you lose your contacts).
If you don't store any personal data on the card, it becomes useless as soon as you either close the account or get a new card from the carrier or MVNO. The card inherently contains 4 pieces of data - the MCC, the MNC, the PUK and the SIM card number - which is how the tower determines which phone is contacting it. (Your phone number is never transmitted.)
There are quite a few apps in Play that can read the contacts on a SIM, and some that can read some of the other data. (You can wrote contacts to the SIM - the other data isn't in writeable storage.)