We analyse consumers' search and purchase decisions on an Internet platform. Using a rich dataset on all adverts posted and transactions made on a major French Internet platform (PriceMinister), we show evidence of substantial price dispersion among adverts for the same product. We also show that consumers do not necessarily choose the cheapest advert available and sometimes even choose an advert that is dominated in price and non-price characteristics (such as seller's reputation) by another available advert. To explain the transactions observed on the platform, we derive and estimate a structural model of sequential directed search where consumers observe all advert prices but have to pay a search cost to see the other advert characteristics. We allow for flexible heterogeneity in consumers' preferences and search costs. After deriving tractable identification conditions for our model, we estimate sets of parameters that can rationalize each transaction. Our model can predict a wide range of consumer search strategies and fits almost all transactions observed in our sample. We find empirical evidence of heterogenous, sometimes positive and substantially large search costs and marginal willingness to pay for advert hedonic characteristics.