by Psi Wavefunction
The complex structure of flagellates. EM section through
a mature flagellate Tetramitus. K1 = kinetosome. Cm =
cytostomal canal. FV = food vacuole. Rz = rhizoplast.
Source.
By default, a membrane-bound entity like a cell should be a spherical, formless blob. However, most cells are not such formless blobs, but rather have adopted one or more forms from a vast repertory of stunningly complex morphologies. To wit, see (and admire) the radiolarians, metazoan neurons, giant parabasalians or the endlessly weird and sophisticated ciliates. Even prokaryotes have a highly complex cellular structure, and are not, as some biochemists are prone to think, mere bags of enzymes. The deeper you venture into the realm of cellular diversity, the more awe-inspiring becomes the cornucopia of cellular structural and morphological variety. Luckily, there is some order to it as there are two fundamental 'genres' of cellular morphology, at least in the protists: flagellates and amoebae. Of course, there are also cysts, but since those are mostly resting stages (being a round ball isn't particularly helpful while feeding or fleeing from predators...) they can be ignored for now.
Cell shape depends on the cytoskeleton. As you know, its two main component systems are actin and tubulin, ignoring the plethora of miscellaneous proteins that are used for various structural jobs. Tubulin makes microtubules, the spindle fibers of mitosis, but is also important for the flagellar apparatus (we've yet to find one composed of actin and probably for good reasons). You also know that actin is a key player in cell motility and morphology in animal systems. It is also heavily involved in endomembrane trafficking within a cell, as well as endo- and exocytosis. If interested, a recent issue of Science has a nice overview of actin in morphogenesis and cell movement.
The role of the cytoskeleton in morphogenesis is much less clearly defined. It depends largely on the species. Plants, for example, rely very heavily on tubulin for morphology, with actin being a minor player. Amoeboid cells are primarily actin-based. In fact, amoeboid cells resort to tubulin largely for spindle formation during mitosis. They hate tubulin about as much as plants hate actin. Actin-based cells don't have to be amorphous; they are still able to achieve complex morphologies. But there is a positive correlation between amoeboid-ness and actin-ness ('actinity'?).
In contrast, flagellates are primarily tubulin-based. Of course, they still use actin for some intracellular work, but the shape depends largely on the whims of their microtubules. Perhaps not relying much on flagella allows the amoeboids to dispense with the microtubule organization pathways, thereby switching to actin. Flagellates, relying heavily on intact tubulin systems, may be less prone to losing their structure. Also, if you're a flagellate, you need shape for a modicum of streamlining. Try swimming around as a formless or floppy blob of some sort! Keep in mind that life at that scale is very different. Viscosity calls the shots when considering unicellular motility. Perhaps being hydrodynamic isn't even as important as simply retaining shape. Otherwise you'd be like a blob of molasses trying to swim through a sea of maple syrup. Not gonna get very far.
Whatever the reason, amoeboid cells tend to have a predominantly actin-based cytoskeleton, flagellates have a penchant for tubulin. Of course, not all organisms are decisive enough to make this commitment, so we've got amoeboflagellates in the middle.
Plenty of other organisms fancy transitioning between being more amoeboid or more flagellate. But few cells actually dispense with flagella and basal bodies altogether, to form them anew when special conditions arise. It's time to introduce one that does.