Plain and spartan both refer to the quality of something that is simple or spare and without any enhancements that provide pleasure, comfort, or ease. Plain suggests something basic, simple, and consisting only of what’s necessary, without pretention or embellishment (a plain room; a plain diet; plain packaging). Spartan suggests a stark simplicity of lifestyle, frugal and minimal to the point of harshness (a spartan diet; spartan conditions). The militarist society of Sparta in ancient Greece, whence we get this adjective, was renowned for its rigorous discipline in every aspect of life. Spartan is often used to describe rooms and furnishings (a spartan room; spartan accommodations).
The more common word trace suggests a mark, sign, or other evidence of the former presence or action of something (trace of blood; traces of a fire). A vestige is such a trace remaining of something that existed in the past, often the very distant past (vestiges of a past era; the only remaining vestige). Trace has a broad range of use, but is most frequently used of a very small amount of substance scientifically detected and analyzed. Vestige is a more evocative word; it can suggest the remainder of something that we can’t otherwise experience or that will never exist again—except in this last vestige.
Both verbs refer to staying or remaining alive despite adverse conditions, circumstances, or events. Survive can suggest living on after something (survived the drought) or living or getting through something very difficult (survived by begging). Subsist is closer to the latter sense of survive. It refers to how a person or animal survives physically, particularly with respect to nutrition and often suggests obtaining the minimum food or nutrients necessary to live (subsist by hunting; subsisted on nuts and berries). Those who are said to subsist on a particular diet, usually eat only one or two types of food, which may be due to scarcity, poverty, or an unbalanced lifestyle (subsisted on a diet of coffee and donuts).