$k$-SAT is defined as the problem of finding models (assignments of either true or false to the variables appearing in the formula) to propositional formulas in Conjunctive Normal Form (CNF). A clause is defined as a disjunction of literals (variables that are either in positive or negative form) and a CNF formula is then just a conjunction of clauses.
Thus, as David Richerby already told you, it is not about formulae that have $k$ clauses.
Take the case you mention, 1-SAT. It is trivial to prove that it can be solved in linear time, $O(k)$ with $k$ being the number of clauses: you just assign value true to the propositions that are positive and false otherwise. It is also possible to prove that a formula is unsatisfiable if it appears in two different (unary) clauses in positive and negative form as you are processing the formula.
Krom showed in 1967 that 2-SAT can be solved in polynomial time.
However, 3-SAT has been known since 1972 to be NP-complete (included in Karp's 21 problems) and there is indeed a lot of work on the (2+p)-SAT where random formulae are generated with clauses of length 2 with probability $(1-p)$ and length 3 with probability $p$ showing that there is a clear transition from polynomial to exponential complexity.