Summary: In this paper, we will discuss the boundary regularity of stationary critical points for the following Cosserat energy functional: \[ \mathrm{Coss}(\phi, R) = \int_\Omega(|R^t\nabla\phi - I_3|^2 + |\nabla R|^p + \langle\phi - x, f\rangle + \langle R, M\rangle)\,dx, \tag{1} \] where \(2\leq p < 3\), \(f\in L^\infty(\Omega, \mathbb{R}^3)\), \(M\in L^\infty(\Omega, SO(3))\), and \(\Omega\subset\mathbb{R}^3\) is a domain with \(C^1\) boundary. Precisely, if \((\phi, R)\in H^1(\Omega, \mathbb{R}^3)\times W^{1, p}(\Omega, SO(3))\) is a stationary critical point of (1) satisfying a certain boundary monotonicity inequality, we show that there exists a closed subset \(\Sigma\subset\partial\Omega\) satisfying the Hausdorff measure \(H^{3 - p}(\Sigma) = 0\) such that \((\phi, R)\in C^{1, \alpha}(\Omega_\delta\setminus\Sigma)\times C^\alpha(\Omega_\delta\setminus\Sigma)\), where \(\Omega_\delta := \{x\in\bar{\Omega}, \mathrm{dist}(x, \partial\Omega) \leq \delta\}\), \(\delta > 0\).