Determining trend, inferring its significance, and detrending are important steps in time series analysis. With the rapid accumulation of complex data in many areas of science and engineering, these operations have become increasingly frequent. To facilitate development of sound detrending algorithms, recently, it is proposed that a trend be defined as a monotonic function (of time) having at most one extremum within a given data span. Using real-world data, we show that such a definition of trend is not all encompassing, and cannot be used to identify significant local trends amidst a global long-term trend. To overcome limitations of existing detrending algorithms, we propose a new local adaptive detrending algorithm, and apply it to determine local and global trends from climatological datasets such as the surface temperature anomalies. Based on a fundamental statistical means of quantifying the significance of local trends of climate variability, we conclude that a recent global surface cooling phase very likely has set in amid the global warming trend.