Enterprises have a growing need to identify relevant tables in data lakes; e.g. tables that are unionable, joinable, or subsets of each other. Tabular neural models can be helpful for such data discovery tasks. In this paper, we present TabSketchFM, a neural tabular model for data discovery over data lakes. First, we propose a novel pre-training sketch-based approach to enhance the effectiveness of data discovery techniques in neural tabular models. Second, to further finetune the pretrained model for several downstream tasks, we develop LakeBench, a collection of 8 benchmarks to help with different data discovery tasks such as finding tasks that are unionable, joinable, or subsets of each other. We then show on these finetuning tasks that TabSketchFM achieves state-of-the art performance compared to existing neural models. Third, we use these finetuned models to search for tables that are unionable, joinable, or can be subsets of each other. Our results demonstrate improvements in F1 scores for search compared to state-of-the-art techniques (even up to 70% improvement in a joinable search benchmark). Finally, we show significant transfer across datasets and tasks establishing that our model can generalize across different tasks over different data lakes