(1) Assuming $T$ depends on amplitude — it does not in SHM. (2) Using $T = 2\pi\sqrt{l/g}$ for a pendulum in a non-inertial frame without replacing $g$ with $g_{\text{eff}}$. (3) Forgetting that cutting a spring changes its spring constant (shorter = stiffer). (4) Confusing phase difference between $v$ and $a$ ($\pi/2$) with that between $x$ and $a$ ($\pi$). (5) Assuming KE and PE oscillate at frequency $\omega$ — they oscillate at $2\omega$. (6) Neglecting to check whether the problem specifies "from mean to extreme" or "from extreme to extreme" when calculating time.