Direct proof of use
The F/A-18E/F Super Hornet is directly documented in the Syrian Civil War through U.S. Navy reporting on Operation Inherent Resolve. On Dec. 29, 2015, Naval Forces Central Command reported that strike aircraft from USS Harry S. Truman flew their first OIR missions and that Truman and Carrier Air Wing 7 were conducting combined combat operations in Iraq and Syria from the Arabian Gulf. The same release identified an F/A-18E Super Hornet assigned to VFA-25 and an F/A-18F Super Hornet assigned to VFA-103 on Truman's flight deck in the deployment context.
A separate Navy release documents a specific air-to-air use inside Syria. On June 18, 2017, the Navy said a Syrian regime Su-22 dropped bombs near Syrian Democratic Forces fighters south of Tabqah and was immediately shot down by a U.S. F/A-18E Super Hornet in collective self-defense of coalition partnered forces.
Sources: Harry S. Truman CSG Launches First OIR Missions, Coalition Defends Partner Forces from Syrian Fighter Jet Attack
Timeline
The supported public record begins in late 2015, after the U.S.-led coalition campaign against ISIS had become part of the wider Syrian war. Truman's first OIR missions placed carrier strike aircraft, including Super Hornets in the embarked air wing, into combat operations covering Iraq and Syria.
The clearest dated Super Hornet engagement in Syria came on June 18, 2017 near Tabqah. The Navy account describes a sequence in which pro-Syrian regime forces attacked SDF-held Ja'Din, coalition aircraft first conducted a show of force, and the Syrian Su-22 was later shot down after dropping bombs near SDF fighters.
Sources: Harry S. Truman CSG Launches First OIR Missions, Coalition Defends Partner Forces from Syrian Fighter Jet Attack
Operational role
In the Syrian Civil War catalog, the Super Hornet appears as a U.S. Navy carrier-based coalition aircraft rather than a system possessed by Syrian forces. The documented uses were carrier strike operations under Operation Inherent Resolve and defensive counter-air action to protect SDF and coalition partner forces.
The Navy's 2015 Truman release ties the aircraft to strike operations in Iraq and Syria against ISIL, while the 2017 Tabqah release identifies the platform, user, opponent aircraft, location, and reason for the engagement. CJTF-OIR history provides the wider campaign context: the Department of Defense formally established Combined Joint Task Force - Operation Inherent Resolve on Oct. 17, 2014 for ongoing military action against ISIS in Iraq and Syria, and later coalition phases supported partner forces in Syria.
Sources: Harry S. Truman CSG Launches First OIR Missions, Coalition Defends Partner Forces from Syrian Fighter Jet Attack, CJTF-OIR History
Coalition and side context
The source-backed side assignment is Syrian Democratic Forces and coalition partners. The 2017 Navy release says the Syrian regime aircraft dropped bombs near SDF fighters and that the F/A-18E acted in collective self-defense of coalition partnered forces. CJTF-OIR history also describes coalition support to partner forces in Syria during the campaign against Da'esh.
The available official sources do not support Syrian possession, transfer to a local force, or routine basing in Syria for the Super Hornet. They support U.S. Navy carrier-based fielding and use over or in relation to Syria during the anti-ISIS campaign and one documented defensive counter-air engagement against a Syrian regime aircraft.
Sources: Coalition Defends Partner Forces from Syrian Fighter Jet Attack, CJTF-OIR History