We investigate carrier estimation (CE) for coherent optical receivers where the received signal is impaired by additive white Gaussian noise, laser phase noise, and frequency offset. Best practical 4-, 8-, and 16-point constellations are identified. A generalized differential encoding rule for signal constellations is presented. Performance of our complex-weighted decision-aided maximum-likelihood (CW-DA-ML) phase noise and frequency offset estimator is analyzed at low signal-to-noise ratio (SNR) and the optimal filter lengths are found. CW-DA-ML CE is put in perspective with respect to two fundamental estimators in the literature: (i) differential frequency estimator followed by block Mth power phase estimator (DiffFE-Mth CE), and (ii) fast Fourier transform based frequency estimator followed by block Mth power phase estimator (FFTbE-Mth CE), in terms of laser linewidth tolerance, frequency estimation range and speed, SNR threshold, and cycle slip probability. CW-DA-ML CE is 2.5 and 10.5 times faster than DiffFE-Mth CE in 4 phase-shift keying and 16 quadrature amplitude modulation signals, respectively, at a 1-dB system penalty for a bit-error rate of 10(-3). Our CE has lower cycle slip probability and transmission overhead than DiffFE-Mth and FFTbE-Mth CE. Hence, our CE is shown to be favourable in pilot-assisted (PA) systems. A PA CW-DA-ML CE is introduced and shown to be robust against time-varying frequency offset with minimal training overhead. Analog-to-digital convertor quantization error on our CE performance is also addressed.