Streaming codes are a class of packet-level erasure codes that are designed with the goal of ensuring recovery in low-latency fashion, of erased packets over a communication network. It is well-known in the streaming code literature, that diagonally embedding codewords of a $[τ+1,τ+1-a]$ Maximum Distance Separable (MDS) code within the packet stream, leads to rate-optimal streaming codes capable of recovering from $a$ arbitrary packet erasures, under a strict decoding delay constraint $τ$. Thus MDS codes are geared towards the efficient handling of the worst-case scenario corresponding to the occurrence of $a$ erasures. In the present paper, we have an increased focus on the efficient handling of the most-frequent erasure patterns. We study streaming codes which in addition to recovering from $a>1$ arbitrary packet erasures under a decoding delay $τ$, have the ability to handle the more common occurrence of a single-packet erasure, while incurring smaller delay $r<τ$. We term these codes as $(a,τ,r)$ locally recoverable streaming codes (LRSCs), since our single-erasure recovery requirement is similar to the requirement of locality in a coded distributed storage system. We characterize the maximum possible rate of an LRSC by presenting rate-optimal constructions for all possible parameters $\{a,τ,r\}$. Although the rate-optimal LRSC construction provided in this paper requires large field size, the construction is explicit. It is also shown that our $(a,τ=a(r+1)-1,r)$ LRSC construction provides the additional guarantee of recovery from the erasure of $h, 1 \leq h \leq a$, packets, with delay $h(r+1)-1$. The construction thus offers graceful degradation in decoding delay with increasing number of erasures.