I know that the complexity class $\mathsf{P}$ has complete problems w.r.t. $\mathsf{NC}$ and $\mathsf{L}$ reductions.Are these two classes the only possible classes of reductions under which $\mathsf{P}$ has complete problems?  Also, what classes of reduction can be used for $\mathsf{NP}$ beside polynomial-time reductions?
Are there complete problems for P and NP under other kinds of reductions?
complexity theory;reductions
Your questions contain a few incorrect or unclear phrases.  Neither complexity class X has Y reduction nor we can use Y reduction for complexity class X makes sense.  In addition, there are at least two definitions known under the name polynomial-time reductions, both of which are used to study NP-completeness: polynomial-time many-one reductions and polynomial-time Turing reductions.  But in this answer, I will ignore the difference between many-one reductions and Turing reductions, and I will focus only on the resource restrictions of reductions, because otherwise the answer will become too long and unfocused.Now I will restate what you might want to ask, and answer them.Are there any definitions of reductions with respect to which NP-completeness can be defined, other than polynomial-time reductions?  Are there any definitions of reductions with respect to which P-completeness can be defined, other than NC reductions and log-space reductions?As Artem and Raphael said, you can define whatever you like.Are there any definitions of reductions actually used to study NP-completeness in the literature, other than polynomial-time reductions?  Are there any definitions of reductions actually used to study P-completeness in the literature, other than NC reductions and log-space reductions?Yes.  For example, Papadimitriou uses log-space reductions throughout his textbook [Pap94], including the definition of NP-completeness.  (Note: in [Pap94], the term L-reduction means something completely different from log-space reduction.)  As for P-completeness, NCk reductions are mentioned in [GHMRSS00].  This is a special case of NC reductions, and more general than log-space reductions for k2.But are they really different notions, or just different definitions for the same notion?Currently, no one knows.  For example, log-space reducibility and polynomial-time reducibility are equivalent if and only if L=P.[GHMRSS00] Raymond Greenlaw, H. James Hoover, Satoru Miyano, Walter L. Ruzzo, Shuji Shiraishi, and Takayoshi Shoudai.  The Parallel Computation Project: Volumes IIII, 2000.  http://www.cs.armstrong.edu/greenlaw/research/PARALLEL/[Pap94] Christos H. Papadimitriou.  Computational Complexity.  Addison-Wesley, 1994.