I cannot comment on 'most common', but I can definitely share several tools and research efforts towards using FPGA for deep-learning. See my survey paper on FPGA-based accelerators for CNN which reviews 75+ recent papers.
Some of these research projects have released their code, such as DNNWeaver. Also, see tools from companies such as Xilinx. Finally, see the code of top FPGA entries at DAC-System-Design-Contest 2018. This contest features embedded system implementation of neural network based object detection for drones.
FPGA development takes time and hence, sometime it may not catch up with the pace of progress in machine learning. Also, unlike tools such as caffe, tensor-flow etc., which are supported by big universities/companies/open-source-communities, a research group alone cannot take their tool/idea to product/maturity-stage. Hence, FPGA development tools are not so numerous as is the case for GPU/CPU. So, I believe understanding "key research ideas" is also important and my answer on state-of-the-art research/academic proposals has some value and others can add different answers.