Let φ ∈ Q[z] be a polynomial of degree d at least two. The associated canonical heightĥ φ is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of φ. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points,ĥ φ is bounded below by a positive constant (depending only on d) times some kind of height of φ itself. In this paper, we provide support for these conjectures in the case d = 3 by computing the set of small height points for several billion cubic polynomials.In that case, we say x is n-periodic; the smallest such positive integer n is called the period of x. More generally, x is preperiodic under φ if there are integers n > m ≥ 0 such that φ n (x) = φ m (x); equivalently, φ m (x) is periodic for some m ≥ 0.In 1950, using the theory of arithmetic heights, Northcott proved that if deg φ ≥ 2, then φ has only finitely many preperiodic points in Q. (In fact, his result applied far more generally, to morphisms of N-dimensional projective space over any number field.) In 1994, motivated by Northcott's result and by analogies to torsion points of elliptic curves (for which uniform bounds were proven by Mazur [13] over Q and by Merel [14] over arbitrary number fields), Morton and Silverman proposed a dynamical Uniform Boundedness Conjecture [17,18]. Their conjecture applied to the same general setting as Northcott's Theorem, but we state it here only for polynomials over Q.Conjecture 1 (Morton, Silverman 1994). For any d ≥ 2, there is a constant M = M(d) such that no polynomial φ ∈ Q[z] of degree d has more than M rational preperiodic points.Thus far only partial results towards Conjecture 1 have been proven. Several authors [17,18,19,21,24] have bounded the period of a rational periodic point in terms of the smallest prime of good reduction (see Definition 1.3). Others [5,12,15,16,22] have proven that polynomials of degree two cannot have rational periodic points of certain periods by studying the set of rational points on an associated dynamical modular curve; see also [23, Section 4.2]. A different method, introduced in [3] and generalized and sharpened in [2], gave (still non-uniform) bounds for the number of preperiodic points by taking into account all primes, including those of bad reduction.