Multichannel frequency estimation with incomplete data and miscellaneous noises arises in array signal processing, modal analysis, wireless communications, and so on. In this paper, we consider maximum-likelihood(-like) optimization methods for frequency estimation in which proper objective functions are adopted subject to observed data patterns and noise types. We propose a universal signal-domain approach to solve the optimization problems by embedding the noiseless multichannel signal of interest into a series of low-rank positive-semidefinite block matrices of Hankel and Toeplitz submatrices and formulating the original parameter-domain optimization problems as equivalent structured matrix recovery problems. The alternating direction method of multipliers (ADMM) is applied to solve the resulting matrix recovery problems in which both subproblems of ADMM are solved in (nearly) closed form. The proposed approach is termed as structured matrix embedding and recovery (StruMER). Extensive numerical simulations are provided to demonstrate that StruMER has improved threshold performances in various challenging scenarios, e.g., limited data, low signal-to-noise ratio, impulsive noise, and closely spaced frequencies, as compared with state-of-the-art methods.