Typically, numerical simulations of the ocean, weather, and climate are coarse, and observations are sparse and gappy. Recently, generative diffusion models have emerged as state-of-the-art tools for image generation and shown promise in various high-dimensional inverse problems. In this work, we apply and benchmark generative diffusion modeling approaches to super-resolution and inference from coarse, sparse, and gappy observations. We apply both guided approaches that minimally adapt a pre-trained unconditional diffusion model and conditional approaches that require training with paired high-resolution and coarse-resolution or observational data. We first show applications to idealized 2D quasi-gesotrophic turbulence on the beta-plane in two dynamical regimes, the eddy regime and the jet regime. Next, we show extensions to inference of surface oceanic flows in the Gulf of Mexico from gappy, noisy observations. Our comprehensive skill metrics include norms of the reconstructed fields, turbulence statistical quantities, quantification of the super-resolved probabilistic ensembles and their errors, and validation of the generated posterior distributions. We also study the sensitivity to tuning parameters such as guidance strength. Our results highlight the trade-offs between ease of implementation, fidelity (sharpness), and cycle-consistency of the diffusion models, and offer practical guidance for deployment in oceanographic and geophysical inverse problems.