We describe a scaffolding approach to the task of coreference resolution that incrementally combines statistical classifiers, each designed for a particular mention type, with rule-based models (for sub-tasks well-matched to determinism). We motivate our design by an oracle-based analysis of errors in a rule-based coreference resolution system, showing that rule-based approaches are poorly suited to tasks that require a large lexical feature space, such as resolving pronominal and common-noun mentions. Our approach combines many advantages: it incrementally builds clusters integrating joint information about entities, uses rules for deterministic phenomena, and integrates rich lexical, syntactic, and semantic features with random forest classifiers well-suited to modeling the complex feature interactions that are known to characterize the coreference task. We demonstrate that all these decisions are important. The resulting system achieves 63.2 F1 on the CoNLL-2012 shared task dataset, outperforming the rule-based starting point by over seven F1 points. Similarly, our system outperforms an equivalent sieve-based approach that relies on logistic regression classifiers instead of random forests by over four F1 points. Lastly, we show that by changing the coreference resolution system from relying on constituent-based syntax to using dependency syntax, which can be generated in linear time, we achieve a runtime speedup of 550 per cent without considerable loss of accuracy.