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We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.
It is proved that for any Banach space X property (β) defined by Rolewicz in [22] implies that both X and X* have the Banach-Saks property. Moreover, in Musielak-Orlicz sequence spaces, criteria for the Banach-Saks property, the near uniform convexity, the uniform Kadec-Klee property and property (H) are given.
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