  The problem of estimating an unknown phase $ \varphi $ using two-level probes
in the presence of unital phase-covariant noise and using finite resources is
investigated. We introduce a simple model in which the phase-imprinting
operation on the probes is realized by a unitary transformation with a randomly
sampled generator. We determine the optimal phase sensitivity in a sequential
estimation protocol, and derive a general (tight-fitting) lower bound. The
sensitivity grows quadratically with the number of applications $ N $ of the
phase-imprinting operation, then attains a maximum at some $ N_\text{opt} $,
and eventually decays to zero. We provide an estimate of $ N_\text{opt} $ in
terms of accessible geometric properties of the noise and illustrate its
usefulness as a guideline for optimizing the estimation protocol. The use of
passive ancillas and of entangled probes in parallel to improve the phase
sensitivity is also considered. We find that multi-probe entanglement may offer
no practical advantage over single-probe coherence if the interrogation at the
output is restricted to measuring local observables.
