Why aren’t any JP-1-type (extreme-narrow-cut) fuels in use today? JP-1 was the first U.S. military standard jet fuel (hence the 1 in the designation), with its specifications issued in 1944; it was an extremely narrow-cut, high-alkane kerosene, not far from what would later become RP-1.  As a result, it had both a very low freezing point (-60C maximum) and a very high flash point, a combination that is the Holy Grail of the aviation-fuel industry.  Its only drawback was that it was hard to produce (since, to quote John D. Clark in Ignition!, “not many refineries in the country could produce such a product with their available equipment and crudes”), and therefore expensive and in chronically short supply, which prompted the development of wide-cut fuels (first the extremely-wide-cut JP-3, followed by the somewhat saner JP-4) and the gradual disappearance of JP-1 (although, judging by mentions in various aircraft accident reports, it apparently was still being produced into the 1970s).

On the other hand, those were 1940s/50s refineries that had difficulty producing JP-1 - and what would be tough beans for a 1940s refinery would be no trouble at all for a 21st-century refinery, thanx to the polymerisation/cracking/hydrogenation/witchcraft/etc. techniques originally developed to turn all the heavier, less-profitable petroleum fractions (and the gaseous, hard-to-store ones) into gold, I mean gasoline, which could easily be adapted to turn even the most recalcitrant crudes into extreme-narrow-cut, JP-1-type fuels (especially given that RP-1 is an even tighter kerosene than JP-1, albeit not by much, and yet we haven’t had any trouble getting ahold of that in the vast quantities needed for launching things into space); so why haven’t we returned to JP-1 and similar fuels, which have freezing points rivalling Jet B combined with flash points that blow away (pun intended) even Jet A?
JP-1 is not in use because it's expensive and there's cheaper ways to deal with flash point and freezing point. 


The navy developed JP-5 for carrier use where a higher flash point is required, and it is easier and cheaper to produce than jp-1
A higher flash point isn't that big a gain in most use cases, it's not as if the fuel is going to instantly burst into flame if it reaches 40&deg;C - with good equipment and safety practices Jet-A(1) is still pretty safe to handle
Anti-gelling additives, tank heaters and recirculators work very well to keep fuel from gelling or freezing. This has proven very effective, BA-38 is the only case I can think of in recent memory where ice in fuel has caused an issue, and that was due to water ice crystals rather than the fuel itself, which didn't freeze. JP-1 wouldn't have helped in that case