This paper presents channel estimation techniques for stacked intelligent metasurface (SIM)-assisted multi-user holographic MIMO (HMIMO) systems. The SIM, which is physically constructed by stacking multiple layers of metasurfaces, has an architecture similar to an artificial neural network and can flexibly manipulate electromagnetic waves. This enables the SIM to achieve HMIMO precoding and combining in the wave domain, reducing hardware cost and energy consumption. In this letter, we investigate the channel estimation problem in SIM-assisted multi-user HMIMO systems. Since the number of antennas at the base station (BS) is much smaller than the number of meta-atoms per layer of the SIM, it is challenging to acquire the channel state information (CSI). To address this, we collect multiple copies of the uplink pilot signals that propagate through the SIM. We leverage the array geometry to identify the subspace that spans arbitrary spatial correlation matrices. Based on partial CSI about the channel statistics, we propose two subspace-based channel estimators. Additionally, we compute the mean square error (MSE) of the proposed channel estimators and optimize the phase shifts of the SIM to minimize the MSE. Numerical results are illustrated to analyze the effectiveness of the proposed channel estimation schemes. The proposed channel estimation techniques include the least squares (LS), the minimum mean square error (MMSE), the reduced-subspace LS (RSLS), and the RSLS relying on isotropic scattering statistics (RSLS-iso). The latter two methods exhibit enhanced estimation accuracy even with limited knowledge of the spatial correlation matrix. Furthermore, the phase shifts of the meta-atoms in the SIM are optimized for improving the performance of the considered channel estimators. Extensive simulation results are illustrated to prove the effectiveness of the proposed channel estimation techniques. The results show that the MMSE and RSLS estimators offer stronger robustness than the LS and RSLS-iso estimators. The proposed techniques significantly improve the channel estimation performance in SIM-assisted HMIMO systems.This paper presents channel estimation techniques for stacked intelligent metasurface (SIM)-assisted multi-user holographic MIMO (HMIMO) systems. The SIM, which is physically constructed by stacking multiple layers of metasurfaces, has an architecture similar to an artificial neural network and can flexibly manipulate electromagnetic waves. This enables the SIM to achieve HMIMO precoding and combining in the wave domain, reducing hardware cost and energy consumption. In this letter, we investigate the channel estimation problem in SIM-assisted multi-user HMIMO systems. Since the number of antennas at the base station (BS) is much smaller than the number of meta-atoms per layer of the SIM, it is challenging to acquire the channel state information (CSI). To address this, we collect multiple copies of the uplink pilot signals that propagate through the SIM. We leverage the array geometry to identify the subspace that spans arbitrary spatial correlation matrices. Based on partial CSI about the channel statistics, we propose two subspace-based channel estimators. Additionally, we compute the mean square error (MSE) of the proposed channel estimators and optimize the phase shifts of the SIM to minimize the MSE. Numerical results are illustrated to analyze the effectiveness of the proposed channel estimation schemes. The proposed channel estimation techniques include the least squares (LS), the minimum mean square error (MMSE), the reduced-subspace LS (RSLS), and the RSLS relying on isotropic scattering statistics (RSLS-iso). The latter two methods exhibit enhanced estimation accuracy even with limited knowledge of the spatial correlation matrix. Furthermore, the phase shifts of the meta-atoms in the SIM are optimized for improving the performance of the considered channel estimators. Extensive simulation results are illustrated to prove the effectiveness of the proposed channel estimation techniques. The results show that the MMSE and RSLS estimators offer stronger robustness than the LS and RSLS-iso estimators. The proposed techniques significantly improve the channel estimation performance in SIM-assisted HMIMO systems.