A soybean cultivar designated S050051 is disclosed. The invention relates to the seeds of soybean cultivar S050051, to the plants of soybean S050051, to plant parts of soybean cultivar S050051 and to methods for producing a soybean plant produced by crossing soybean cultivar S050051 with itself or with another soybean cultivar. The invention also relates to methods for producing a soybean plant containing in its genetic material one or more transgenes and to the transgenic soybean plants and plant parts produced by those methods. This invention also relates to soybean cultivars or breeding cultivars and plant parts derived from soybean cultivar S050051, to methods for producing other soybean cultivars, lines or plant parts derived from soybean cultivar S050051 and to the soybean plants, varieties, and their parts derived from use of those methods. The invention further relates to hybrid soybean seeds, plants and plant parts produced by crossing cultivar S050051 with another soybean cultivar.