Owning shares is a means of supporting a club financially. It does give the shareholder a stake in the club, albeit small, and while it may have certain benefits such as the right to attend meetings it's value is limited, it's very unlikely to be an investment that will make a financial return unless an interested party seeks to buy a large number of shares from ordinary fans.
However if you band together with other shareholders, say through a supporters trust then those small individual shareholdings may in total be big enough to have some influence on the club's decisions and future direction.
I'll probably buy some shares if the opportunity arises, partly to feel I've got a tangible personal stake in Argyle but also to support the Chairman in his efforts to further develop the club.