To copy iPhone SMS to computer:
Step 1: Download iPhone SMS Transfer free trial version here, then install it. You also need iTunes above 7.3 installed.
Step 2: Connect iPhone to your computer.
Step 3: Launch iPhone SMS Transfer, the SMS contacts info will display as shown in your iPhone screen automatically as fig 1. Click on certain contact, all SMS records from him/she will display as fig 2 shown.

Fig 1 (Click Here Enlarge)

Fig 2 (Click Here Enlarge)
Step 4 Option 1: In fig 1 situation, you can click button "Copy ALL SMS" to copy sms from all iPhone contact, then select options:
1: Choose File Type: back up to TXT file or ANTS file;
2: Choose File Path: where to store the backup file, do nothing if you choose default path;
3: Advanced Option: if you choose ANTS format in step 1, you can add a password to protect the file.
Click OK Button to finish the Copy. See fig 3.

Fig 3 (Click Here Enlarge)
Step 4 Option 2: In fig 2 situation, click button "Copy SMS From who" to copy sms from a single contact, select options:
1: Choose File Type: You can only select TXT file format in single contact sms transfer;
2: Choose File Path: where to store the backup file, do nothing if you choose default path;
3: Advanced Option: Disabled in single contact sms transfer.
Click OK Button to finish the Copy. See fig 4.

Fig 4 (Click Here Enlarge)
Tips: What is ANTS file?
Ants file format is developed for storing iPhone SMS data on PC. The ANTS file with file extension ".ants" can be opened by this iPhone SMS Transfer tool by double click or drag it to the software window, then the file display as Fig 2 above.
ANTS file format support strong-encryption password protection method, so ANTS file is safe for personal privacy if you provide a password to the file.