Is there a common fetal position?

Yes. Arms, legs & thighs are all flexed. Body and head are curled forwards like a ball.
Cephalic. The most common fetal position is cephalic (head-down). Other possible positions: breech (bottom-down), oblique (unstable lie) and transverse (back-up or back-down). These change many times daily, but less so near term. Only cephalic fetuses can be born safely vaginally; all others should probably be rotated to cephalic or delivered via cesarean section.