In reference to the fictional superhero Spider-Man, the expression spider sense (used in the singular and in the plural) denotes a supernatural ability or power to perceive things beyond the normal range of human senses, especially impending danger.

The earliest occurrence that I have found of the expression spider sense used in reference to Spider-Man is from Spider-Man vs. the Chameleon!, in the first issue of The Amazing Spider-Man (New York: Non-Pareil Publishing Corporation, March 1963), written by Stan Lee (Stanley Martin Lieber – 1922-2018) [page 6, panel 1]—image: SpiderFan.org:

SPIDER-MAN HAS THE POWERS AND INSTINCTS OF A SPIDER! SO I WILL SEND HIM A MESSAGE THAT ONLY HIS SPIDER SENSES WILL BE ABLE TO PICK UP!

However, the expression spider sense originally denoted an apparent ability to sense or intuit the presence of nearby spiders.

This expression occurred, for example, in Chapter XVI, entitled Spider-Sense and Cat-Sense, of Great and Small Things (London: Methuen and Co. Ltd., 1923), by the British zoologist Edwin Ray Lankester (1847-1929) [pages 138 & 139]:

THERE is at the present day a more general disposition than was the case thirty or forty years ago to dabble in “occultism”—to seriously relate and discuss stories and theories as to ghosts, divination, second sight, and mysterious inherited memories of long-past ages. This change of attitude is not accounted for by any discoveries of a scientific nature tending to give support to popular superstitions or to so-called “occultism.” The fact is that there is a distinct lowering in the standard of veracity and sound common sense which not long ago characterized the best English journalism. Newspapers, formerly written for serious men, now not unfrequently cater for those who desire tit-bits of scandal, and also for lovers of pseudoscientific mysteries and medical quackery decked out with sham learning and airs of profundity.
Among the mysteries thus offered to the contemplation of the public is one which has been dubbed the “spider-sense.” It is related that there are persons who not only have an extreme and unaccountable dread and dislike of spiders, but that some of them are brought into a strange state of nervous agitation by the proximity of a spider, and may even faint in consequence. Not only is this extreme nervous disturbance reported, but it is further stated that such individuals are thus affected by the presence of a spider in the same room with them, even when it is not seen by the sufferer nor its presence suspected by others. The susceptible individuals have insisted on a search being made for the unseen spider, and it is stated by witnesses present on such occasions that after hunting about in corners and among shelves the offending spider has been discovered and ejected, whereupon the agitated individual (a lady in one case) has recovered serenity. On this basis we are seriously, and with an air of exceptional learning, asked to admit the existence of a peculiar sense—not that of sight, of hearing, of smell, of taste, or of the various kinds grouped as “touch” (enumerated in Chapter X). This peculiar “sense” is assumed to be possessed by some individuals and not by others, and to enable those individuals to recognize the presence of a spider when other persons cannot do so. It is proposed to call this the “spider-sense,” and by the more elaborately phantastic of these wonder-mongers the manifestation is compared to recorded cases in which a cat takes the place of the spider, and we are gravely assured that there is a “cat-sense” which is similar to but, of course, not identical with, the “spider-sense.” Both “spider-sense” and “cat-sense” are, it seems probable, a variety of non-sense!

This use of spider sense seems to have originated in British newspapers, chiefly in The Times of London, in 1914. The earliest occurrences that I have found of this expression are as follows, in chronological order:

1-: From The Times (London, England) of Wednesday 18^th March 1914 [page 11, column 5]:

WOMAN’S SPIDER SENSE.
INSTINCTIVE DETECTION OF INSECT’S PRESENCE.
A DOCTOR’S EXPERIENCE.

A medical correspondent writes:—
A case of a most remarkable kind has just come to my notice. A few weeks ago, while on a visit to the country, I met a young man who informed me that his wife possessed a most extraordinary sixth sense in regard to spiders. She could detect the presence of a spider in any room she happened to be living in, without having seen the insect or, indeed, if one may so put it, without having any reason to suppose that it was there. The discovery was accompanied by violent sickness, malaise, and even debility, but all these symptoms at once passed away when the spider was caught and removed from the room.
I had heard of such cases, and recollected having read a story in one of the medical papers some years ago in which a somewhat similar state of affairs was described, yet found it hard to believe that actual detection of so small an insect could occur as it were “by instinct.” However, my doubts were set at rest a few nights later when the lady referred to joined her husband at the house where we were staying. In the middle of the night my new acquaintance came to my room and asked me to attend his wife, who had become very unwell. He added, “She declares there is a spider in the bed-room, but I cannot find one this time.” I followed him and found his wife in a state which suggested sudden collapse. She was very pale, with a feeble pulse and rapid breathing. She declared that she felt “dreadfully sick” and that she was absolutely certain that there was a spider somewhere in the room.
So insistent was she on this point that, to humour her, but without in the least believing her story, her husband and I lit a candle and searched every nook and cranny of the room. We found nothing, and were about to give up the rather ridiculous pursuit, when the patient suddenly announced that she “had a feeling” that the spider was upon the mantelpiece. We looked there and had satisfied ourselves that she was quite mistaken when it occurred to me to lift the edge of the flounce surrounding the woodwork. As I did so a large black spider ran quickly along the cloth towards a hole in the wood and disappeared. The husband and I looked at one another and I signed to him to afford no indication of what had occurred. But just then a sigh of relief from the bed, accompanied by the remark, “At last you have found it,” proved to us the futility of our precaution. The sixth sense had not failed.
Within about half an hour the patient was quite well again, and, on being assured that the hole in the woodwork was stopped up, fell asleep peacefully.

2-: From one of the unconnected paragraphs making up the column Notes and News, published in The Birmingham Daily Mail (Birmingham, Warwickshire, England) of Thursday 19^th March 1914 [page 4, column 2]:

A medical journal has just discovered that there is such a thing as a “spider sense.” It is recognised that there is an instinctive antipathy to cats among certain people, of whom Lord Roberts is popularly supposed to be one; and now it is seriously declared that certain people, ladies especially, have the power of feeling the presence of a spider in a room, and that spiders make such people feel so ill that they cannot stay in the same room. If so, what, one may ask, is the explanation? Will the evolutionists regard it as a racial memory of the spider and the fly incident? Are we to accept it as proof of the transmigration theory that these people so endowed were in some previous existence living in the bodies of flies? Or is the whole suggestion a hoax?

3-: From The Times (London, England) of Friday 20^th March 1914 [page 11, column 3]:

THE SPIDER SENSE.
TO THE EDITOR OF THE TIMES.

Sir,—The incident mentioned by your correspondent of a lady being influenced by a spider reminds me of a very similar case in my own family. My uncle’s wife was always put in a state of terror when a spider was near, in spite of it being hidden from her. I have heard my uncle say how he would have to hunt for them in the bedroom at night. He tried at first to ridicule the whole affair and treat the matter as a form of hysteria, but so persistent was his wife that she could only be calmed by his finding the spider hidden behind some curtain pole or between the mattresses. I have known her not being able to sit in church for the same cause.
I can speak from my own knowledge, having often seen her in terrible distress until the spider was caught or removed.
Faithfully yours,
JOHN PLACE, M.I.Mech.E.

4-: From The Times (London, England) of Saturday 21^st March 1914 [page 11, column 4]:

A WOMAN’S SIXTH SENSE.
SURVIVAL FROM PRIMITIVE TIMES.
(FROM A CORRESPONDENT.)

Two striking instances of what may be called a sixth sense have found themselves recently and oddly united in the same copy of The Times. The one is frankly headed, “A Woman’s Spider Sense.” It gives an attested account of a lady somehow endowed with a strange subconsciousness of spiders, however hidden—habitually collapsing when they are there and reviving whenever and however they are removed. The other—a mere straw in the daily whirlpool of news—relates how a sister’s dream—legally verified—discovered the secret site of her brother’s suicide. Both of these cases raise fascinating problems and open out avenues to discussion, and even discovery. It may well be that the spider-sense and the dream-sense are in essence affinities, both different dimensions of a subtle “sixth sense”; both at once premonitions and preservatives.
[&c.]