The impending realization of scalable quantum computers will have a significant impact on today's security infrastructure. With the advent of powerful quantum computers public key cryptographic schemes will become vulnerable to Shor's quantum algorithm, undermining the security current communications systems. Post-quantum (or quantum-resistant) cryptography is an active research area, endeavoring to develop novel and quantum resistant public key cryptography. Amongst the various classes of quantum-resistant cryptography schemes, lattice-based cryptography is emerging as one of the most viable options. Its efficient implementation on software and on commodity hardware has already been shown to compete and even excel the performance of current classical security public-key schemes. This work discusses the next step in terms of their practical deployment, i.e., addressing the physical security of lattice-based cryptographic implementations. We survey the state-of-the-art in terms of side channel attacks (SCA), both invasive and passive attacks, and proposed countermeasures. Although the weaknesses exposed have led to countermeasures for these schemes, the cost, practicality and effectiveness of these on multiple implementation platforms, however, remains under-studied.