Select the contact from the contact list -> click the menu icon (the very left icon at the bottom of your phone below "Samsung" -> hit Edit -> scroll down to "Ringtone"
Please note, I called tech support AND went down to the local Verizon store. THERE IS NO "assign ringtone" in the "Edit" feature of a Contact. The rep at the store, who has a Fascinate, also did not have this feature on his phone BUT he did suggest getting two free apps from the market. He stated that "ringdroid" would find the default ringtones..... NOT. I have yet to find a simple solution and will either create my own mp3 ringtones or return the Fascinate and get the Droid X. I'm sick of calling Tech support every day since I got the phone on Wednesday and also making a trip to the store where I purchased it and NOT given a correct or workable solution. I had the Motorola Droid and just wanted the bigger screen, not bigger headaches!
Found another solution. In the google search engine type "Samsung Fascinate assign ringtones", find the link about assigning "custom" ringtones.
Hope this helps, but what a hassle!
I bet all your contacts are Google/Gmail imported right? I had the same problems when I got the phone. I used to have a Droid so when I switched over my contacts where synced through Gmail....and thats where the problem is at...and here's the bad news: you can assigned different ringtones to your contacts but they have to be IN PHONE CONTACTS...meaning that they must be saved directly into the phone(the old fashion way) not sync'd from another source. To prove my point try this: open up your contacts and click "add new" fill in the usual info but after you enter in the phone number dont hit save but keep scrolling down and guess what you'll see......yep, "ringtone" Weird but thats just how this phone is. After finding this out I had to stop gmail from sync'ing my contacts, wrote down all my contacts, erased the contacts that were on the phone, and began the long process of typing in all my contacts into the phone. have fun....