Geologists commonly purport that successions of strata show one or more thickening and/or thinning upward trends, often prompting colleagues to argue that the `trends' are subjectively identified, unproven or nonexistent. Parametric and randomization tests are proposed to evaluate the null hypothesis of random succession against a variety of alternative postulates of trend. In place of test statistics in vogue that merely compare each bed thickness with that of the beds immediately above and below it, test statistics based on Kendall's S and Tau that make sequence-wide (or subsequence-wide) comparisons of bed thicknesses are advocated. The test statistic used and the exact form of the test depends on the alternative model considered: against the alternative of a single thickening (and/or thinning) upward trend, Kendall's S or equivalently Kendall's Tau are recommended. These statistics make pair-wise comparisons of beds, comparing bed thicknesses with their positions in the vertical sequence. Against the alternative of trends in g subsequences recognized a-priori, e.g. those separated by breaks such as thick sequences of hemipelagic shale, test statistics proposed include: the weighted sum of the g Tau coefficients calculated for the individual subsequences (if subsequences are alleged to be all thickening or all thinning upward), and the weighted sum of the absolute value, or square, of the Tau coefficients (if subsequences are alleged to include both thickening and thinning upward patterns). Tests can indicate that a sequence has one or more subsequences which are nonrandom, but it will not indicate which. To test each subsequence for significance, test each of g subsequences at a level of significance = α/g, thus achieving an overall, sequence-wide, level of significance = α. Against the alternative g subsequences recognized post-hoc, i.e. purely on the basis of observed thickness patterns, a family of test statistics are proposed, each equal to the maximum value of the appropriate test statistic (defined for subsequences recognized a-priori) that is attainable by partitioning the total sequence of beds into 1, 2,…. up to g subsequences. Both same-type and mixed subsequences alternatives arise. Each test proposed is applied to several different sequences, mostly turbidites.