We report on the experimental characterisation and theoretical prediction of intensity fluctuations for ultrafast fibre amplifiers. We formulate a theoretical model with which the intensity noise of a Yb-doped fiber amplifier can be predicted with high accuracy, taking into account seed and pump noise, as well as generation of amplified spontaneous emission. Transfer of pump and seed signal modulations to the amplified output during fibre amplification is investigated thoroughly. Our model enables design and optimisation of fiber amplifiers with regards to their intensity noise performance. As a route to passively decreasing the noise imparted by multi-mode diodes in cladding-pumped amplifiers, we evaluate the impact of using multiple, low-power pump diodes versus a single, high-power diode in terms of the noise performance. We use this gathered intuition on intensity noise to build a low noise fibre interferometer that is able to detect sub-5 nm vibrations for biological experiments.