Because of your use of the word "can", I assume that you mean "at most" in each case. As ymbirtt pointed out, it's probably best just to write it in English. Here is how I would write it with logical formulas. The advantage of this way is that it is easy to generalize to other situations.
It's easier to express (1) as "at least one is nonzero" than "at most two are zero" and similarly it's easier to express (2) as "at least two are nonzero" than "at most one is zero", so we will do it this way.
$a \ne 0 \vee b \ne 0 \vee c \ne 0$
$(a \ne 0 \wedge b \ne 0) \vee (a \ne 0 \wedge c \ne 0) \vee (b \ne 0 \wedge c \ne 0)$
$(a \ne 0 \wedge b \ne 0 \wedge c \ne 0)$
Although the parentheses in case (3) aren't necessary, I included them so you can see the pattern. Each case has been expressed as a disjunction ("or") of conjunctions ("and" statements,) so it is said to be in disjunctive normal form.
As you can see, if we try to do this with more variables $a$, $b$, $c$, $d$,..., the statements will get very long. This is why it is better to write it in English: formal logic can't concisely express "at least $n$ many of the statements...are true" or "at most $n$ many of the statements...are true" (not without bringing in set theory, anyway.)