Elegance, grace, dignity are comparable only when they denote an impressive beauty of form, appearance, or behavior.
Elegance is used in reference to persons chiefly when their grooming, their clothes, and the way they wear them are specifically considered; it then often implies fashionableness and good taste, but it stresses perfection of detail and exquisiteness or, sometimes, overexquisiteness (as in materials, lines, and ornamentation).
When used in reference to such things as the furnishings of a home, the details of a dinner, or a literary style, the term also implies the perfection and propriety in detail that indicate excellence of taste, a nice selective instinct, and often a restrained luxuriousness.
Grace is more commonly applied to what is inward and native than to what is outward and acquired, especially when used in reference to persons; it always suggests a quality or a harmonious combination of qualities that gives aesthetic pleasure through a natural or simple beauty such as is shown in suppleness or rhythm of movement, in clean-flowing lines or contours, or in spontaneity and felicitousness of manner, mood, expression, or style.
Dignity applies to what compels respect and honor. The term often suggests stateliness, majesty, and elevation of character or style as the compelling cause. Very frequently in modern use the term suggests the compulsion of intrinsic worth or merit apart from any superficial characteristics that give it external beauty.