We present three high finesse tunable monolithic fiber Fabry-Perot cavities (FFPCs) with high passive mechanical stability. The fiber mirrors are fixed inside slotted glass ferrules, which guarantee an inherent alignment of the resonators. An attached piezoelectric element enables fast tuning of the FFPC resonance frequency over the entire free-spectral range for two of the designs. Stable locking of the cavity resonance is achieved for sub-Hertz feedback bandwidths, demonstrating the high passive stability. At the other limit, locking bandwidths up to tens of kilohertz, close to the first mechanical resonance, can be obtained. The root-mean-square frequency fluctuations are suppressed down to ∼2% of the cavity linewidth. Over a wide frequency range, the frequency noise is dominated by the thermal noise limit of the system’s mechanical resonances. The demonstrated small footprint devices can be used advantageously in a broad range of applications like cavity-based sensing techniques, optical filters or quantum light-matter interfaces.