Pressboards are commonly used as insulating materials employed in electrical connections of transformers. Pressboards are typically made from vegetable fibers, which contain cellulose. The proper operation of power transformer depends mainly on constant monitoring of insulation materials against failure. Due to the complex and close structure of power transformers, it is very challenging task to detect failure and hence possible location of degradation of pressboard internally. Generated discharge signals may result in breakdown of system insulation and system failure. In this study, the investigation of insulation degradation is fulfilled by analyzing discharge signals and simultaneously produced acoustic signals during discharges. For this purpose, a test setup is used for investigating discharge signals of pressboard samples under different electrical stresses. This paper proposes monofractal and multifractal analysis of discharge and acoustic signals of pressboards. The Higuchi's method is an effective monofractal analysis tool for measurement of fractal dimension of self-affine signals, which is proposed for online monitoring of discharge signals of pressboards. In order to investigate obtained discharge signals with accelerated fluctuations effectively, multifractal detrended fluctuation analysis is proposed for these signals, which exhibit nonlinear behavior.