Microscopic magnetic resonance elastography (μMRE) is a phase contrast based imaging technique that is capable of mapping the acoustic shear waves resulting from low amplitude cyclic displacement in tissue-like materials. This new technique has proven successful in imaging gel phantoms mimicking soft biological tissues with shear moduli ranging from 0.7 to 40 kPa. The 4-dimensional (4D) spatial-temporal shear wave vector can be measured, which in turn can be used to identify material properties with high spatial resolution. Experiments were conducted using 5 and 10 mm RF saddle coils in the 10 mm vertical imaging bore of an 11.74 Tesla magnet. The field-of-view ranged from 4 to 14 mm, with in plane resolution up to 34 μm x 34 μm and slice thickness up to 100 μm using shear wave excitation of 550 to 580 Hz. In this study, the capability and constraints of μMRE are investigated. The constraints include the range of measured shear moduli, excitation frequency, and minimum physical sample volume. Applications investigated include: 1) late-stage frog oocytes with typical diameter from 1 to 1.5 mm; and 2) tissue engineered constructs at different growth stages. Mesenchymal stem cells (MSCs) extracted from bone marrow can serve as progenitor cells that differentiate into specific types of tissues such as bone, adipose tissue, cartilage and muscle. μMRE can monitor the growth of such tissues and evaluate their mechanical properties. Also, a silicon-based tissue phantom material (CF-11-2188, Nusil Technologies) is tested in order to address challenges associated with excitation frequency and the dispersive nature of the media.