Due to the high vulnerability of SRAM-based FPGAs in single-event upsets (SEUs), effective fault tolerant soft processor architectures must be considered when we use FPGAs to build embedded systems for critical applications. In the past, the detection of symptoms of soft errors in the behavior of microprocessors has been used for the implementation of low-budget error detection techniques, instead of costly hardware redundancy techniques. To enable the development of such low-cost error detection techniques for FPGA soft processors, we propose an in-depth analysis of the symptoms of SEUs in the FPGA configuration memory. To this end, we present a flexible fault injection platform based on an open-source CAD framework (Rapid-Smith) for the soft error sensitivity analysis of soft processors in Xilinx SRAM-based FPGAs. Our platform supports the estimation of soft error sensitivity per configuration bit/frame, processor component and benchmark. The fault injection is performed on-chip by a dedicated microcontroller which also monitors processor behavior to identify specific symptoms as consequences of soft errors. The performed analysis showed that these symptoms can be used to build an efficient, low-cost error detection scheme. The proposed platform is demonstrated through an extensive fault injection campaign in the Leon3 soft processor.