This paper presents different state feedback approaches of finite control set model predictive control (FCS-MPC) applied to a grid-connected voltage-source converter (VSC) with an LCL filter. Besides converter-side current feedback, two multivariable control approaches and line-side current control are introduced and compared based on theoretical and experimental evaluation. As the LCL filter introduces an additional resonance frequency to the system, the use of different active damping (AD) methods in combination with FCS-MPC is discussed. Furthermore, practical control implementation issues are discussed. The presented methods reveal the great potential, high dynamic performance, and flexibility of FCS-MPC, enabling multivariable control as well as both reduced switching losses and low harmonic current distortion at the same time. Eventually, the feasibility of the theoretical control concepts is shown in a laboratory environment.