Most real numbers

Most real numbers are...
- irrational : $\mathbb{Q} \subseteq \mathbb{R}$ has measure 0
- uncomputable : a number is computable $\Leftrightarrow$ exists turing machine which outputs it, but there are only countably many Turing machines
- normal (the digits of their $n$-ary expansion are uniformly distributed) : Borel's Theorem
- members of a (particular) meagre set : see Fat Cantor Sets
- transcedental : Each non-transcedental number is the root of a polynomial, there are countably many polynomials, and each polynomial has finitely many roots
- S-numbers of type 1 : proven by Sprindzhuk, at least according to this book, p. 86 (I couldn't find the actual proof). Also see this Wikipedia page.
- not definable : there are only countably many formulas in ZFC

Most real numbers have...
- a corresponding subset of $\mathbb{N}$ with positive upper density : most real numbers are normal, so by definition the upper density of their corresponding set is $\frac{1}{2}$
- a continued fraction expansion $a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \dots}}$ with $\lim_{n \to \infty} (a_1a_2\dots a_n)^\frac{1}{n} = K_0 \approx 2.68545$ : see Khinchin's Constant (contribution from Nick Mahdavi)
- a continued fraction expansion $a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \dots}}$ with $\lim_{n \to \infty} a_n^\frac{1}{n} = e^{\pi^2/(12 \ln{2})} \approx 3.27582$ : see Lévy's constant
- an irationality exponent of 2 : a corollary of the Borel-Cantelli lemma (also see this paper)