We experimentally examine the Poiseuille flow of viscoelastic fluids over wavy surfaces. Five precision microfabricated flow channels are utilized, each of depth 2d= 400μm, spanwise width4= 10d and with a sinusoidal undulation of amplitude A=d/20 on one of the spanwise walls. The undulation wavelength λ is varied between each of the channels, providing dimensionless channel depths α in the range 0.2π≤α= 2πd/λ≤3.2π. Nine viscoelastic polymer solutions are formulated, spanning more than four orders in elasticity number El and are tested in the wavy channels over a wide range of Reynolds and Weissenberg numbers. Flow velocimetry is used to observe and measure the resulting flow patterns. Perturbations to the Poiseuille base flow caused by the wavy surfaces are quantified by the depth of their penetration P into the flow domain. Consistent with theoretical predictions made for wavy plane-Couette flow [J. Page and T. A. Zaki, “Viscoelastic shear flowover a wavy surface,” J. Fluid Mech.801, 392–429 (2016)], we observe three distinct flow regimes (“shallow elastic,” “deep elastic” and “transcritical”) that can be assembled into a “phase diagram” spanned by two dimensionless parameters:α and the depth of the theoretically predicted critical layerΣ∼√El. Our results provide the first experimental verification of this phase diagram and thus constitute strong evidence for the existence of the predicted critical layer. In the inertio-elastic transcritical regime, a surprising amplification of the perturbation occurs at the critical layer, strongly influencing P. These effects are of likely importance in widespread inertio-elastic flows in pipes and channels, such as in polymer turbulent drag reduction.