The one-dimensional Galerkin-truncated Burgers equation, with both dissipation and noise terms included, is studied using spectral methods. When the truncation-scale Reynolds number R min is varied, from very small values to order 1 values, the scale-dependent correlation time τ ( k ) is shown to follow the expected crossover from the short-distance τ ( k ) ∼ k − 2 Edwards–Wilkinson scaling to the universal long-distance Kardar–Parisi–Zhang scaling τ ( k ) ∼ k − 3 / 2 . In the inviscid limit, R min → ∞ , we show that the system displays another crossover to the Galerkin-truncated inviscid-Burgers regime that admits thermalized solutions with τ ( k ) ∼ k − 1 . The scaling forms of the time-correlation functions are shown to follow the known analytical laws and the skewness and excess kurtosis of the interface increments distributions are characterized. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.