Flexible metal-organic frameworks (MOFs) can show remarkable properties that can be useful for various gas separation and storage applications. However, due to the relatively small number of data compared to those of rigid MOFs, the computational design of flexible MOFs with the desired properties has been challenging. In this study, we propose novel methods for designing MOFs that possess potential flexibility and discover new flexible MOFs capable of exhibiting an S-shaped isotherm, indicating a high gas working capacity. Employing our methods, we constructed a comprehensive database comprising MOFs with potential flexibility, which we subsequently screened using molecular simulations to identify optimal MOFs with high CH4 and H-2 working capacity and flexibility. Specifically, we have identified five optimal candidates that may exhibit S-shaped isotherms with exceptional CH4 working capacity. Furthermore, we have determined that these optimal candidates have wide-ranging applicability for other gases such as H-2. In particular, one of the candidate MOFs (named Co+E11) shows a theoretical working capacity of 214.5 v(STP)/v for CH4 (298 K, 5.8-65 bar) and 45.0 g/L for H-2 (77 K, 5-100 bar), surpassing the current world record of 208 v(STP)/v for CH4 and 37.2 g/L for H-2. Hence, we demonstrate the expansion of the data-driven MOF space into the realm of flexible MOFs and propose new synthesizable candidates for gas storage applications.