When does the convolution of $2$ signals equal zero?
Time-domain convolution is frequency-domain multiplication. If at all frequencies at least one of the signals is zero-valued in frequency domain, then the convolution of the two signals will be zero-valued at all frequencies, and at all times. Except for a zero signal, no finite-length signal has a continuous run of frequency domain zeros, so your choices are limited to infinite-length signals such as periodic signals with no coinciding non-zero harmonics and ideal lowpass–highpass filter impulse response pairs.