Fluctuations in the large-scale structure of the Universe contain significant information about cosmological physics, but are modulated in survey datasets by various observational effects. Building on existing literature, we provide a general treatment of how fluctuation power spectra are modified by a position-dependent selection function, noise, weighting, smoothing, pixelization and discretization. Our work has relevance for the spatial power spectrum analysis of galaxy surveys with spectroscopic or accurate photometric redshifts, and radio intensity-mapping surveys of the sky brightness temperature including generic noise, telescope beams and pixelization. We consider the auto-power spectrum of a field, the cross-power spectrum between two fields and the multipoles of these power spectra with respect to a curved sky, deriving the corresponding power spectrum models, estimators, errors and optimal weights. We note that "FKP weights" for individual tracers do not in general provide the optimal weights when measuring the cross-power spectrum. We validate our models using mock datasets drawn from N-body simulations †. Our treatment should be useful for modelling and studying cosmological fluctuation fields in observed and simulated datasets.The power spectrum of the large-scale structure of the Universe -and its dependence on scale, redshift and direction -contains significant information about the composition of the Universe and the cosmological physics governing the growth of structure with time. Modern cosmological surveys can trace this large-scale structure over large volumes, by mapping the individual redshift-space positions of galaxies or quasars, the cumulative brightness temperature of spectral emission in a region of sky using intensity mapping in radio wavebands, or the spectral absorption of background light by intervening matter.One of the central problems in cosmological analysis is to relate these measured fluctuations in probes of large-scale structure, which are modulated by various observational effects and analysis approximations, to the underlying matter power spectrum which encodes the important cosmological information. Relevant observational effects may include a variation in the mean background level of the fields as a function of position (the survey selection function or mask), noise due to the sampling of discrete objects or in the measured brightness temperature, or smoothing of the fields in the mapping process due to the telescope resolution. Analysis approximations may involve the pixelization or gridding technique employed, and wide-angle corrections to the local plane-parallel approximation.Moreover, we may also utilize the cross-correlation between two different observed fields which trace the same underlying matter fluctuations. Such a multi-tracer analysis offers several benefits: (1) uncorrelated noise components in the two fields will bias the amplitude of their auto-power spectra, but not their cross-power spectrum; (2) an additive systematic component afflicting one of the...