We study the time and space complexity of randomized Test-And-Set (TAS) implementations from atomic read/write registers in asynchronous shared memory models with $n$ processes. We present an adaptive TAS algorithm with an expected (individual) step complexity of $O(\log^\ast k)$, for contention $k$, against the oblivious adversary, improving a previous (non-adaptive) upper bound of $O(\log\log n)$ (Alistarh and Aspnes, 2011). We also present a modified version of the adaptive RatRace TAS algorithm (Alistarh et al., 2010), which improves the space complexity from $O(n^3)$ to $O(n)$, while maintaining logarithmic expected step complexity against the adaptive adversary. We complement this upper bound with an $\Omega(\log n)$ lower bound on the space complexity of any TAS algorithm that has the nondeterministic solo-termination property (which is a weaker progress condition than wait-freedom). No non-trivial lower bounds on the space requirements of TAS were known prior to this work.