In the early phase, the number of new radio (NR) users is considerably low compared to the long-term evolution (LTE) users. Hence, allocating the LTE spectrum to a small number of NR users will have a detrimental effect on the LTE users. To overcome this backdrop, dynamic spectrum sharing (DSS) between LTE and NR has become an emerging technology where both systems can coexist on the same LTE spectrum. However, the most critical factor for LTE and NR coexistence is to share the spectrum resources of LTE systems with data and control signals necessary to operate NR systems subject to not impacting LTE operations. In line with this, a comprehensive understanding of DSS for LTE and NR coexistence taking into account diversified concerns is presented in this paper. First, an overview of the technological background of essential signals and channels of both LTE and NR systems to enable DSS for their coexistence is given. Numerous deployment options and potential design considerations of DSS are then discussed. The application of DSS to both multicast-broadcast single-frequency network (MBSFN) and non-MBSFN LTE subframes under NR with and without synchronization signal/ physical broadcast channels (PBCH) block (SSB) transmissions is examined. Further, the third-generation partnership project (3GPP) standardization activities towards DSS in its different releases are discussed, numerous use cases are highlighted, and the significance and limitations of DSS are identified. To provide additional insights, the relative comparison of potential aspects of several DSS concerns including, deployment options, design considerations, DSS applications, and 3GPP standardization efforts, is carried out. Finally, a summary of key lessons learned is provided in terms of numerous facts and figures.