Another large family of distributions that contains both $U[0,1]$ inputs (leading to the usual Plancherel measure) and $U\{1,2,\ldots,d\}$ (as in Ryan O'Donnell's answer) as special cases is the following. Let $(\alpha, \beta, \gamma)$ be an element of the *Thoma simplex*, i.e., a triple where
$\gamma\in [0,1]$ and $\alpha$ and $\beta$ are vectors
$$ \alpha = (\alpha_1, \alpha_2, \ldots) \ \ \textrm{ with }\alpha_1\ge\alpha_2\ge \ldots \ge 0,
$$
$$ \beta = (\beta_1, \beta_2, \ldots) \ \ \textrm{ with }\beta_1\ge\beta_2\ge \ldots \ge 0,
$$
and such that
$$
\sum_n \alpha_n + \sum_n \beta_n + \gamma = 1.
$$
We can associate with $(\alpha, \beta, \gamma)$ a probability distribution of a random variable $X$ that is supported on $(0,1) \cup \{1,2,3,\ldots\} \cup \{-1,-2,-3,\ldots\}$, defined by
$$ \mathbb{P}(X \in (a,b)) = \gamma (b-a) \qquad (0<a<b<1),
$$
$$ \mathbb{P}(X = n) = \alpha_n \qquad (n=1,2,\ldots),
$$
$$ \mathbb{P}(X = -n) = \beta_n \qquad (n=1,2,\ldots).
$$
If we now take a sequence of i.i.d. random variables $X_1,X_2,\ldots$ with the same distribution as $X$ and apply the RSK algorithm to them, the resulting random infinite Young tableau has a very natural probability distribution studied by Vershik and Kerov, and supposedly related to the representation theory of the infinite symmetric group (a subject I know nothing about).

Note, however, that requires a modified version of RSK in which the rules for the insertion tableau are different: the rows and the columns of the insertion tableau are weakly increasing, but it is not allowed to have the same negative integer twice in the same row and it is not allowed to have the same positive integer twice in the same column.

This result was proved by Sergei V. Kerov and Anatol M. Vershik (SIAM J. Algebraic Discrete Methods, 7(1):116–124, 1986; available here)

Some additional consequences are discussed in the paper "Robinson-Schensted-Knuth algorithm, jeu de taquin, and Kerov-Vershik measures on infinite tableaux" by Piotr Śniady (SIAM J. Discrete Math. 28 (2014), 598–630; available at these links: journal version, arXiv version.)