The Global Navigation Satellite Systems (GNSS) are increasingly suffering from interferences, such as coming from jammers and spoofers, and their performance is still modest in challenging urban and indoor scenarios. Therefore, there are efforts worldwide to develop complementary positioning, navigation, and timing (PNT) solutions. One of such complementary methods under current research is the so-called LEO-PNT, namely PNT solutions based on Low Earth Orbit (LEO) satellites, and in particular on small-sized or miniaturized satellites. Such satellites have low-to-moderate costs of building, launching, and maintenance. Several challenges are to be overcome when designing a new LEO-PNT solution, concerning all three satellite segments: the signal-in-space (SIS) or space segment, the ground segment, and the user/receiver segment. This paper presents a survey of the SIS design challenges under the inherent constraints of wireless-channel propagation impairments as well as some design recommendations for SIS features. We address different constellation types, achievable coverage limits, and Geometric Dilution of Precision (GDOP) bounds, as well as achievable carrier-to-noise ratios (CNR) under a realistic wireless channel model, based on a Matlab Quadriga simulator. We also discuss several optimization criteria regarding LEO-PNT SIS design, by taking into account the tradeoff between a low cost/low number of satellites on orbit, on one hand and a sufficient coverage and good CNR for PNT purpose on the other hand.