Formal and ceremonious, when used to describe people rather than events, refer to conduct that observes the manners and conventions of behavior expected in certain settings or at serious or important events. Someone whose manner is formal is polite and dignified. When the situation doesn’t call for this, formal can connote stiffness or detachment: Why was she so formal with us? Ceremonious suggests someone who takes great care to observe the manners appropriate to solemn or important occasions: ceremonious manner; a ceremonious bow. As with formal, it can carry a negative connotation of excessive or exaggerated formality when the occasion doesn’t call for it. In novels from the past, many a farewell was taken with a slightly overdone ceremonious bow.
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.