The structure of food webs, complex networks of interspecies feeding interactions, plays a crucial role in ecosystem resilience and function, and understanding food web structure remains a central problem in ecology. Previous studies have shown that key features of empirical food webs can be reproduced by low-dimensional "niche" models. Here we examine the form and variability of food web niche structure by fitting a probabilistic niche model to 37 empirical food webs, a much larger number of food webs than used in previous studies. The model relaxes previous assumptions about parameter distributions and hierarchy and returns parameter estimates for each species in each web. The model significantly outperforms previous niche model variants and also performs well for several webs where a body-size-based niche model performs poorly, implying that traits other than body size are important in structuring these webs' niche space. Parameter estimates frequently violate previous models' assumptions: in 19 of 37 webs, parameter values are not significantly hierarchical, 32 of 37 webs have nonuniform niche value distributions, and 15 of 37 webs lack a correlation between niche width and niche position. Extending the model to a two-dimensional niche space yields networks with a mixture of one- and two-dimensional niches and provides a significantly better fit for webs with a large number of species and links. These results confirm that food webs are strongly niche-structured but reveal substantial variation in the form of the niche structuring, a result with fundamental implications for ecosystem resilience and function.