Although the title only mentions slope bounds, this paper, joint with A. Deopurkar, really investigates the divisor theory of moduli spaces of trigonal curves. The main result illustrates the importance of two special divisors in the geometry of moduli spaces of trigonal curves (suitably compactified). In even genera, the special divisor under consideration is simply the classical Maroni divisor, parametrizing trigonal curves with imbalanced Tschirnhausen bundle -- we compute the divisor class of the closure of this divisor in the space of twisted stable maps. More interestingly, in odd genera we illustrate the importance of the Tangency divisor -- the closure of the locus of trigonal curves which are tangent to the directrix in their relative canonical surface scrolls. This divisor provides a sharp upper bound on the slope of any sweeping curve in the moduli space of trigonal curves. It would be interesting to see how this Tangency divisor (and its significance) might generalize to, say, stacks of tetragonal or pentagonal curves -- this is something I mean to return to one day.