The online option for Suffolk County case search is available at http://www.suffolkcountyny.gov/Departments/CountyClerk/CourtActionsandCo.... This is the link that will give you access to the court minutes. If you need in depth information pertaining to a judicial matter, your best option would be to launch an 'in person' case search either with one of the tribunals or the clerk of court's office.
Ideally, if you have some information on the legal matter that you are investigating such as the Suffolk County court dockets number assigned to it, the inquiry can be conducted through the tribunal in which the matter was fought out. Also, it will be that much easier and faster to find the judicial records pertaining to the litigation.
Of course, if it is a criminal matter that you are inquiring about, you will also be asked to provide the name of the defendant and his birth date. Even though the Supreme and the Municipal Courts will accept requests for name based case searches that are connected to criminal history, these will promptly be forwarded to the OCA and not handled by the court itself.
If you are after civil court documents, these can also be sought from the Supreme Court. However, you need to understand that this tribunal will only handle civil actions in which the amount under dispute is above $15,000. For other cases, you will have to connect with the Municipal Court.
If you are averse to the prospect of running from one court to the other, you could always try the Suffolk County clerk of court's office. This agency is known to keep all judicial records, regardless of the case type. In fact, they manage the central repository of court dockets for Suffolk County, NY.
The clerk of court's department can be visited at 310 Center Dr, Riverhead, NY 11901-3392 from 9 AM to 5 PM. The Supreme Court also works out of the same location. However, for the Municipal tribunal, you will have to drive down to N County Complex Bldg 158, Veteran's Memorial Highway, Hauppauge, NY 11788-0099.