The nautilus. The chambered nautilus shell is a logarithmic spiral, and logarithmic spirals are genuinely common in nature, because that is the shape a structure takes when it grows by adding to one end without changing shape. It is not a golden spiral. When Clement Falbo measured shells in the collection of the California Academy of Sciences he found ratios near 1.33, not 1.618; a later study of eighty shells in the Smithsonian collection put the mean nearer 1.31. Expressed as growth per turn, the nautilus roughly triples its radius in a full revolution, where a golden spiral multiplies its radius by about 6.85. Overlay the two curves and they do not resemble each other.
The Parthenon. George Markowsky's Misconceptions about the Golden Ratio, published in 1992, showed that the golden rectangles habitually drawn over photographs of the Parthenon do not fit: parts of the building fall outside the rectangle, and because the structure is not rectangular and has no unambiguous edges, a measurer can pick reference points that yield almost any ratio wanted. Defenders reply that other reference points, taken from the base of the steps rather than the columns, do give phi. That is the problem rather than the rebuttal. A ratio that depends on which edge you choose is not evidence.
The Great Pyramid. The same objection applies with more force. There are a great many available lengths, the original casing is largely gone, and ratios close to phi, to pi and to several other constants can all be extracted from the same monument by choosing differently. No Egyptian text mentions the ratio.
The two spirals. The familiar diagram of quarter-circle arcs drawn through a chain of Fibonacci squares is not a golden spiral. It is piecewise circular, each arc struck from a different centre, and its curvature jumps at every join, where a true logarithmic spiral has a single centre and curvature that varies smoothly. It converges on the golden spiral as the squares grow, so it is a good approximation, but it is not the same curve. The two are conflated almost universally, including in most published illustrations of the golden spiral.