Apps that run directly on the SIM card are relatively unknown in the US. We (in Canada) normally download apps that run on the phone's OS, like Android or iOS apps. Running apps directly on the SIM card is very unlikely.
I played around with a TurboSim for a while too back when I was testing out a SIM card 'firewall' that would block the carrier programmed SIM from responding to OTA updates or type-0 stealth SMS and other bad things http://www.bladox.com/ then phones with wifi that didn't require a SIM came out.
There are also USSD menus which I hadn't heard of before going to Asia. It was used by carriers there to provide an alternative to voice menus to topup and buy addons. In India there is a company providing access to Facebook over it:
This really isn't that accurate anymore. Verizon Wireless, which is the largest carrier in the US right now, uses sim cards in all of their LTE devices. Behind them is AT&T which of course is completely GSM and T-Mobile. Sprint, well... they're there.