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The nuclear structure of $^{29}$Mg was probed using the $d$($^{28}$Mg,$p$)$^{29}$Mg transfer reaction to populate its single-particle states. The ISOLDE facility at CERN provided a 9.473$\cdot A$ beam of $^{28}$Mg which was directed at a deuterated target within the ISOLDE solenoidal spectrometer. Exploiting the kinematic advantages of this technique allowed most states up to 5 MeV to be resolved, and angular distributions of the reaction cross section to be obtained. The DWBA code DWUCK5 was used to obtain spectroscopic factors for these states. Additionally, some higher-lying excited states were identified, and their possible properties were proposed. Theoretical calculations in this region broadly reproduced the observed behaviour in $^{29}$Mg, as well as matching trends from other nuclides in the $N = 17$ isotones. These calculations indicate that the nucleon-nucleon interaction between protons and neutrons, with the tensor interaction as a key component, is driving the evolution of shell structure, with the filling of the $\pi d_{5/2}$ orbital of particular importance in this region. Finite geometry effects also play an important role in this evolution as the $p$ orbitals approach the neutron-separation threshold, and the energy spacing between them reduces. The information gained from the nuclear structure will help to continue refining these interactions, and provide a valuable benchmark for nuclear-structure studies around the border of the island of inversion. Similar experiments to study $^{30}$Al and $^{31}$Mg carried out recently will help to further the understanding of nuclear structure in this exotic region of the nuclear chart.