Spin waves are investigated in Yttrium Iron Garnet (YIG) waveguides with a thickness of 39 nm and widths ranging down to 50 nm, i.e., with aspect ratios thickness over width approaching unity, using Brillouin Light Scattering spectroscopy. The experimental results are verified by a semi-analytical theory and micromagnetic simulations. A critical width is found, below which the exchange interaction suppresses the dipolar pinning phenomenon. This changes the quantization criterion for the spin-wave eigenmodes and results in a pronounced modification of the spin-wave characteristics. The presented semi-analytical theory allows for the calculation of spin-wave mode profiles and dispersion relations in nano-structures.Spin waves and their quanta, magnons, typically feature frequencies in the GHz to THz range and wavelengths in the micrometer to nanometer range. They are envisioned for the design of faster and smaller next generational information processing devices where information is carried by magnons instead of electrons [1][2][3][4][5][6][7][8][9]. In the past, spin-wave modes in thin films or rather planar waveguides with thickness-towidth aspect ratios ar = h/w << 1 have been studied. Such thin waveguides demonstrate the effect of "dipolar pinning" at the lateral edges, and for its theoretical description the thin strip approximation was developed, in which only pinning of the much-larger-in-amplitude dynamic in-plane magnetization component is taken into account [10][11][12][13][14][15]. The recent progress in fabrication technology leads to the development of nanoscopic magnetic devices in which the width w and the thickness h become comparable [16][17][18][19][20][21][22][23]. The description of such waveguides is beyond the thin strip model of effective pinning, because the scale of nonuniformity of the dynamic dipolar fields, which is described as "effective dipolar boundary conditions", becomes comparable to the waveguide width. Additionally, both, in-plane and out-of-plane dynamic magnetization components, become involved in the effective dipolar pinning, as they become of comparable amplitude.Thus, a more general model should be developed and verified experimentally. In addition, such nanoscopic feature sizes imply that the spin-wave modes bear a strong exchange character, since the widths of the structures are now comparable to the exchange length [24]. A proper description of the spin-wave eigenmodes in nanoscopic strips which considers the influence of the exchange interaction, as well as the shape of the structure, is fundamental for the field of magnonics.In this Letter, we discuss the evolution of the frequencies and profiles of the spin-wave modes in nanoscopic waveguides where the aspect ratio ar evolves from the thin film case ar → 0 to a rectangular bar with ar → 1. Yttrium Iron Garnet (YIG) waveguides with a thickness of 39 nm and widths ranging down to 50 nm are fabricated and the quasi-ferromagnetic resonance (quasi-FMR) frequencies within them are measured using microfocused Brillouin Ligh...