Polar codes are a recent family of error-correcting codes with a number of desirable characteristics. Their disruptive nature is illustrated by their rapid adoption in the $5^{th}$-generation mobile-communication standard, where they are used to protect control messages. In this work, we describe a two-stage system tasked with identifying the location of control messages that consists of a detection and selection stage followed by a decoding one. The first stage spurs the need for polar-code detection algorithms with variable effort to balance complexity between the two stages. We illustrate this idea of variable effort for multiple detection algorithms aimed at the first stage. We propose three novel blind detection methods based on belief-propagation decoding inspired by early-stopping criteria. Then we show how their reliability improves with the number of decoding iterations to highlight the possible tradeoffs between accuracy and complexity. Additionally, we show similar tradeoffs for a detection method from previous work. In a setup where only one block encoded with the polar code of interest is present among many other blocks, our results notably show that, depending on the complexity budget, a variable number of undesirable blocks can be dismissed while achieving a missed-detection rate in line with the block-error rate of a complex decoding algorithm.