In general, multimode fiber sensors are based on light amplitude detection. But the single-mode fibers are required in interferometry systems such Fabry-Perot, Mac- Zehnder interferometer.
Direct detection usually means that the photon is absorbed, and then the resulting electron hole pairs are separated by a field. For example the internal field of a pn junction. Either single mode or multimode fibers can be used to deliver the light. Usually single and mutimode are terms used to describe light sources or waveguides. I haven't seen it used when talking about detectors as much. Hypothetically you can construct systems that can distinguish between different modes, in practice this can get complicated to do, especially if there are numerous modes.
Singlemode and multimode fibers can be used for all types of direct detection (IM-DD systems are currently the most used). When you utilize multimode fiber, modal dispersion must be considered.