An earthenware shard (that  was broken off from a larger utensil and can hold the minimum shiur of liquid for it to be susceptible to tumah)

that cannot stand

due to [the weight of] its handle

or if it had a protrusion

and the protrusion tilts it,

is tahor.

If the handle was removed

or the protrusion broke off,

it is tahor (because once tahor, a fragment cannot become tamei);

Rabbi Yehuda rules it is tamei.

A barrel that is missing a piece (such that it cannot hold liquids when upright)

and it can hold [items] in its walls,

or it split [such that it is now shaped]

like two troughs,

Rabbi Yehuda rules is tahor (as he holds that a barrel is not used to store items while on its side),

and the Sages rule is tamei (as they hold a barrel would occasionally be used to store items while on its side).

פרק ד משנה ב

A barrel that has cracked

and cannot be moved

while [holding] a half-kav (~0.6 to 1.2 liters) of dried figs

is tahor.

A ruined vessel (made of earthenware, as described later this still has some, but not all, of its original utility) that (further) cracked

and cannot hold liquids,

even though

it can hold food,

is tahor

as there are no remnants of remnants (i.e., once ruined, any further ruin removes such from the status of being a vessel that is susceptible to tumah).