In index modulation schemes, information bits are conveyed through indexing of transmission entities such as antennas, subcarriers, times slots, precoders, subarrays, and radio frequency (RF) mirrors. Index modulation schemes are attractive for their advantages such as good performance, high rates, and hardware simplicity. This paper focuses on index modulation schemes in which multiple transmission entities, namely, {\em antennas}, {\em time slots}, and {\em RF mirrors}, are indexed {\em simultaneously}. Recognizing that such multidimensional index modulation schemes encourage sparsity in their transmit signal vectors, we propose efficient signal detection schemes that use compressive sensing based reconstruction algorithms. Results show that, for a given rate, improved performance is achieved when the number of indexed transmission entities is increased. We also explore indexing opportunities in {\em load modulation}, which is a modulation scheme that offers power efficiency and reduced RF hardware complexity advantages in multiantenna systems. Results show that indexing space and time in load modulated multiantenna systems can achieve improved performance.