  Ideals in the ring of power series in three variables can be classified based
on algebra structures on their minimal free resolutions. The classification is
incomplete in the sense that it remains open which algebra structures actually
occur; this realizability question was formally raised by Avramov in 2012. We
discuss the outcomes of an experiment performed to shed light on Avramov's
question: Using the computer algebra system Macaulay2, we classify a billion
randomly generated ideals and build a database with examples of ideals of all
classes realized in the experiment. Based on the outcomes, we discuss the
status of recent conjectures that relate to the realizability question.
