Aqueous lyotropic liquid crystals (LLCs) comprise a class of ordered morphologies formed by self-assembly of amphiphiles in water. LLCs assume a variety of concentration- and temperature-dependent structures including lamellae (bilayers), bicontinuous networks, hexagonally-packed cylindrical micelles, and spherical micelles packed on a lattice. Typically, LLC sphere packings include high-symmetry body-centered cubic (BCC), face-centered cubic (FCC), and hexagonally closest-packed (HCP) structures. Recently, a giant tetragonal σ phase containing 30 micelles of five different sizes was discovered in the aqueous LLC self-assembly of dianionic alkylphosphonate surfactants. The σ phase belongs to a class of tetrahedrally close-packed structures called Frank-Kasper (FK) phases, which possess ≥ 7 particles of two or more types situated at 12-, 14-, 15-, or 16-fold coordination environments in low-symmetry unit cells. Ubiquitous in intermetallic alloys, FK phases have been recapitulated in other soft materials including dendritic thermotropic liquid crystals, giant-shape surfactants, and block polymers. The observation of these complex morphologies across different soft material classes stabilized by varying non-covalent interactions begs the question of universality in the principles that govern FK phase formation. The formation of the σ phase in LLCs of ionic surfactants was rationalized based on maximizing counterion-mediated intermicellar cohesion, while minimizing expensive local variations in headgroup-counterion solvation. However, molecular design principles guiding FK phase selection in LLCs are lacking. FK phases are periodic approximants of dodecagonal quasicrystals (DDQCs), structures which possess 12-fold rotational symmetry yet lack translational symmetry. DDQCs have been observed in self-assembled micelles of neat, neutral amphiphiles in regions of phase space adjacent to FK morphologies. However, quasiperiodic ordering of micelles in LLC self-assembly is surprisingly unknown given the pervasiveness of the periodic approximants. This thesis elucidates the amphiphile structural motifs that stabilize FK phases and related DDQCs in aqueous LLCs. We first establish the molecular design criteria for the formation of σ phases in ionic amphiphiles by investigating the LLC phase behavior of alkylmalonate dianionic surfactant analogous to the alkylphosphonate amphiphiles. FK phase formation was observed to depend on the nature of the counterions and length of the alkyl tail. Using real-space electron density reconstructions, we find that the preference for local micellar symmetry in the σ phase is dictated by the extent of headgroup-counterion association. We next report the formation of a well-ordered DDQC in oil-swollen micelles of alkylphosphonate surfactants, and we use high-resolution small-angle X-ray scattering data to determine the space group symmetry of this quasiperiodic structure. The formation of the DDQC was contingent on the sample-processing protocols employed, indicating the metastability of this mesophase. We further illustrate the non-specific nature of FK phase formation in soft materials by the discovery of a σ phase on self-assembly of hydrated non-ionic polyethylene-block-poly(ethylene oxide) surfactants. For the hydroxyl terminated surfactant, access to the σ phase depends on sample thermal history, indicating its metastability with respect to the A15 structure. Finally, the hydroxyl end-group of the amphiphile was synthetically modified with ionic and strongly H-bonding moieties. We find that strongly interacting terminal groups provide increased temperature- and composition-windows of σ phase stability. Moreover, cationically-terminated oligomers surprisingly self-assembled into a DDQC. These findings are rationalized based on the drive to minimize local variations in intramicellar chain-chain interactions, while maximizing intermicellar cohesion. These fundamental studies of the thermodynamics of micellar morphologies in solvated amphiphiles provide insights into the general underlying principles which stabilize these complex packings of soft reconfigurable particles.