Accurate short-term electricity price forecasts are essential to all electricity market participants. Generation companies adopt price forecasts to hedge generation shortage risks; load serving entities use price forecasts to purchase energy with low cost; and trading companies utilize price forecasts to arbitrage between markets. Currently, researches on point forecast mainly focus on exploring periodic patterns of electricity price in time domain. However, frequency domain enables us to identify more information within price data to facilitate forecast. Besides, price spike forecast has not been fully studied in the existing works. Therefore, we propose a short-term electricity price forecast framework that analyzes price data in frequency domain and consider price spike predictions. First, the variational mode decomposition is adopted to decompose price data into multiple band-limited modes. Then, the extended discrete Fourier transform is used to transform the decomposed price mode into frequency domain and perform normal price forecasts. In addition, we utilize the enhanced structure preserving oversampling and synthetic minority oversampling technique to oversample price spike cases to improve price spike forecast accuracy. In addition to point forecasts, market participants also need probabilistic forecasts to quantify prediction uncertainties. However, there are several shortcomings within current researches. Although wide prediction intervals satisfy reliability requirement, the over-width intervals incur market participants to derive conservative decisions. Besides, although electricity price data follow heteroscedasticity distribution, to reduce computation burden, many researchers assume that price data follow normal distribution. Therefore, to handle the above-mentioned deficiencies, we propose an optimal prediction interval method. 1) By considering both reliability and sharpness, we ensure the prediction interval has a narrow width without sacrificing reliability. 2) To avoid distribution assumptions, we utilize the quantile regression to estimate the bounds of prediction intervals. 3) Exploiting the versatile abilities, the extreme learning machine method is adopted to forecast prediction intervals. The effectiveness of proposed point and probabilistic forecast methods are justified by using actual price data from various electricity markets. Comparing with the predictions derived from other researches, numerical results show that our methods could provide accurate and stable forecast results under different market situations.