1> Analytical - Black Scholes formula for Vanilla European options, Digitals.
These valuations are just an "interpolation" of traded options. We interpolate the implied volatility from the traded points on the implied volatility surface.
There is no modeling assumption involved here.
Market uses this formula for implementing the "interpolation".
Scope for going wrong using this method (compared to market participants) is very limited. Most issues arise only when volatility surface needs to be extrapolated.
Quanto Europeans also use this method frequently, although this is more than just an interpolation. Modeling assumptions are implicit - (i) Lognormal terminal distributions of the underlying and the FX rate, (ii) correlation input to the analytical formula is often not an "interpolated" value from traded options; hence needs further modeling assumptions.
2> Numerical integration - all payoffs which depend only on the terminal distribution of the underlying, e.g. self-quanto options
3> PDE Method (Finite difference or Trees) - for path dependent payoffs with single underliers (low dimensionality).
4> Monte Carlo - for path dependent payoffs with higher dimensionality e.g. basket options or rates products with multiple Forward Libor rates as the underliers.