Is every regular/context free langauge decidable in LogSpace?
For regular languages, we have 
    $$ \text{REG = DSPACE}(O(1)) = \text{NSPACE}(O(1))$$
where REG is the class of regular languages. This can be easily seen from the equivalent formalisms of regular languages. A language is a regular language iff it is the language accepted by a deterministic finite automaton. A language is a regular language iff it is the language accepted by a nondeterministic finite automaton.

In fact, $\text{REG = DSPACE}(o(\log \log n))$. That is, $Ω(\log\log n)$ space is required to recognize any non-regular language [1]. 

For context-free languages, please see the answer by Yuval Filmus.