A signalling NaN does not occur as a result of an arithmetic computation. A signalling NaN occurs when you load a bad value from memory into a floating-point register that happens to has the same bit pattern as a signalling NaN. IEEE 754 requires that in such a situation the processor should raise an exception. It should then convert the signalling NaN to a quiet NaN so the lifetime of a signalling NaN will be brief.
A signalling NaN's binary representation has a full exponent and its 2 most significant bits are 1