A search through this site yielded plenty of examples where a space is locally compact but not sigma compact, but not the other way around. The fact that there is a term of sigma locally compact suggests that there are counter-examples, but I am having trouble finding one. I would appreciate any such counter-examples.
Also I was wondering about the validity if the following condition would imply local compactness: Let $X$ be a topological space such that $X=\cup_{n=1}^\infty K_n$ where $K_n$ is compact. If $\{K_n \}_{n=1}^\infty$ is locally finite, then $X$ is locally compact. This argument seems to work even if we have a general locally finite compact cover (every set is compact) and not just countable. Under such conditions can we conclude local compactness?
For the first part: $\mathbb Q$ with the usual topology is sigma compact but not locally compact.