Even leaving aside concerns of cognitive plausibility, incremental parsing is appealing for applications such as speech recognition and machine translation because it could allow the incorporation of syntactic features into the decoding process without blowing up the search space. Nevertheless, incremental parsing is often associated with greedy parsing decisions and intolerable loss of accuracy. Would the use of lexicalized grammars provide a new perspective on incremental parsing? In this paper we explore incremental left-to-right dependency parsing using a lexicalized grammatical formalism that works with lexical categories (supertags) and a small set of combinatory operators. A strictly incremental parser would conduct only a single pass over the input, use no lookahead and make only local decisions at every word. We show that such a parser suffers heavy loss of accuracy. Instead, we explore the utility of a two-pass approach that incrementally builds a dependency structure by first assigning a supertag to every input word and then selecting an incremental operator that allows assembling every supertag with the dependency structure built thus far to its left. We instantiate this idea in different models that allow a trade-off between aspects of full incrementality and performance, and explore the differences between these models empirically. Our exploration shows that a semi-incremental (two-pass), linear-time parser that employs fixed and limited look-ahead exhibits an appealing balance between the efficiency advantages of incrementality and the achieved accuracy. Surprisingly, taking local or global decisions matters very little for the accuracy of this linear-time parser. Such a parser fits seamlessly with the currently dominant finite-state decoders for machine translation.