There is growing interest in the role of structural variants (SVs) as drivers of local adaptation and speciation. From a biodiversity genomics perspective, the characterisation of genome-wide SVs provides an exciting opportunity to complement single nucleotide polymorphisms (SNPs). However, little is known about the impacts of SV discovery and genotyping strategies on the characterisation of genome-wide SV diversity within and among populations. Here, we explore a near whole-species resequence dataset, and long-read sequence data for a subset of highly represented individuals in the critically endangered kākāpō (Strigops habroptilus). We demonstrate that even when using a highly contiguous reference genome, different discovery and genotyping strategies can significantly impact the type, size and location of SVs characterised genome-wide. Further, we found that the mean number of SVs in each of two kākāpō lineages differed both within and across generations. These combined results suggest that genome-wide characterisation of SVs remains challenging at the population-scale. We are optimistic that increased accessibility to long-read sequencing and advancements in bioinformatic approaches including multi-reference approaches like genome graphs will alleviate at least some of the challenges associated with resolving SV characteristics below the species level. In the meantime, we address caveats, highlight considerations, and provide recommendations for the characterization of genome-wide SVs in biodiversity genomic research.