Yes, you can read the information on your SIM card. There are USB SIM card readers and software available for purchase - the Dekart SIM Card reader is one example. Once you have purchased a SIM card reader, simply place your SIM card inside, and insert the reader into a spare USB drive on your computer. It should show up as an extra drive on your computer. To get the information off of the SIM, follow the instructions that came with your reader to run the software.
Most programs will give you an option to "Extract" the information, or a "Start" button. After extracting the information, you will see what is stored on your SIM, including your current and deleted SMS messages, saved contacts, your card identification number and the name of your service provider.