In the context of autonomous driving in urban environments accurate and reliable information about the vehicle motion is crucial. This article presents a multi-modal sensor fusion scheme that, based on standard production car sensors and an inertial measurement unit, estimates the three-dimensional vehicle velocity and attitude angles (pitch and roll). Moreover, in order to enhance the estimation accuracy, the scheme simultaneously estimates the gyroscope and accelerometer biases. The approach relies on a state-affine representation of a kinematic model with an additional measurement equation based on a single-track model. The sensor fusion scheme is built upon a recently proposed adaptive estimator, which allows a direct consideration of model uncertainties and sensor noise. In order to provide accurate estimates during collision avoidance manoeuvres, a measurement covariance adaptation is introduced, which reduces the influence of the single-track model when its information is superfluous. A validation using experimental data demonstrates the effectiveness of the method during both regular urban drives and collision avoidance manoeuvres.