Elementary methods

1. Finite automata (possibly nondeterministic, with empty transitions).
2. Regular expressions.
3. Right (or Left, but not both) linear equations, like $X = KX + L$ where $K$ and $L$ are regular.
4. Operations preserving regular languages (Boolean operations, product, star, shuffle, morphisms, inverses of morphisms, reversal, etc.)
5. Recognized by a finite monoid.

Logical methods (often used in formal verification)

6. Monadic second order logic (Büchi's theorem).
7. Linear temporal logic (Kamp's theorem).
8. Rabin's tree theorem (Monadic second order logic with two successors). Very powerful.

Advanced methods

9. Sophisticated pumping lemmas. See for instance
   [1] J. Jaffe, A necessary and sufficient pumping lemma for regular languages, Sigact News - SIGACT 10 (1978) 48-49.
   [2] A. Ehrenfeucht, R. Parikh, and G. Rozenberg, Pumping lemmas for regular sets, SIAM J. Comput. 10 (1981), 536-541.
   [3] S. Varricchio, A pumping condition for regular sets, SIAM J. Comput. 26 (1997) 764-771.

10. Well quasi orders. See
    [4] W. Bucher, A. Ehrenfeucht, D. Haussler, On total regulators generated by derivation relations, Theor. Comput. Sci. 40 (1985) 131–148.
    [5] M. Kunz, Regular Solutions of Language Inequalities and Well Quasi-orders.

11. Support of $\mathbb{N}$-rational series.

12. Algebraic methods based on Transductions (see also Operations preserving regular languages).

Elementary methods

1. Finite automata (possibly nondeterministic, with empty transitions).
2. Regular expressions.
3. Right (or Left, but not both) linear equations, like $X = KX + L$ where $K$ and $L$ are regular.
4. Regular (Type 3) grammar.
5. Operations preserving regular languages (Boolean operations, product, star, shuffle, morphisms, inverses of morphisms, reversal, etc.)
6. Recognized by a finite monoid.

Logical methods (often used in formal verification)

7. Monadic second order logic (Büchi's theorem).
8. Linear temporal logic (Kamp's theorem).
9. Rabin's tree theorem (Monadic second order logic with two successors). Very powerful.

Advanced methods

10. Sophisticated pumping lemmas. See for instance
    [1] J. Jaffe, A necessary and sufficient pumping lemma for regular languages, Sigact News - SIGACT 10 (1978) 48-49.
    [2] A. Ehrenfeucht, R. Parikh, and G. Rozenberg, Pumping lemmas for regular sets, SIAM J. Comput. 10 (1981), 536-541.
    [3] S. Varricchio, A pumping condition for regular sets, SIAM J. Comput. 26 (1997) 764-771.

11. Well quasi orders. See
    [4] W. Bucher, A. Ehrenfeucht, D. Haussler, On total regulators generated by derivation relations, Theor. Comput. Sci. 40 (1985) 131–148.
    [5] M. Kunz, Regular Solutions of Language Inequalities and Well Quasi-orders.

12. Support of $\mathbb{N}$-rational series.

13. Algebraic methods based on Transductions (see also Operations preserving regular languages).