English grammar tells us that:
I haven’t got a clue.
I don’t have a clue.
are the correct ways to express negative possession. However, there is the rarer, older British sounding:
I haven’t a clue.
Notice that a superlative phrase is common to give emphasis:
I haven’t the slightest idea how he works.
Here are the search results from iWeb corpus for:
_VH _XX _A _N
1 HAVEN’T A CLUE 1948
Nicholas, I haven’t a clue what this is all about,
but your taste in champagne is excellent, as always.
2 HADN’T A CLUE 401
3 HASN’T A CLUE 270
4 HAVE NOT A CLUE 265
5 HAVEN’T THE TIME 236
6 HAVE NOT THE SPIRIT 132
7 HAVE NOT THE POWER 121
8 HAD NOT A CLUE 118
9 HAS NOT THE POWER 111n
10 HAVE NOT THE TIME 108
11 HAD NOT THE POWER 104
12 HAS NOT THE RIGHT 99
13 HAVE NOT THE MEANS 92
14 HAD NOT THE COURAGE 91
15 HAD NOT THE STRENGTH 68
16 HAVE NOT THE LAW 57
17 HAVE NOT THE RIGHT 55
18 HAVE NOT THE COURAGE 54
19 HAVEN’T A CHANCE 53
20 HAD NOT THE HEART 52
21 HAD NOT THE MEANS 52
22 HAD NOT A WORD 52
23 HAVEN’T THE MONEY 52
24 HATH NOT A JEW 46
25 HAVEN’T THE HEART 46
26 HAS NOT A CLUE 45
27 HAS NOT THE TIME 44
Unfortunately, he has not the time to hear everyone sing.
28 HAS NOT A WORD 44
29 HASN’T A CHANCE 43
30 HAVE NOT A LOT 43
31 HADN’T THE TIME 42
32 HAS NOT THE SPIRIT 42
33 HAVEN’T A HOPE 41
34 HAVE NOT A DOUBT 41
35 HAVE NOT THE STRENGTH 40
36 HAD NOT THE TIME 38
37 HAVING NOT THE SPIRIT 38
38 HADN’T A CARE 35
39 HAD NOT A CARE 35
40 HAS NOT A RIGHT 34
41 HAVEN’T THE PATIENCE 34
42 HAVEN’T THE STRENGTH 34
43 HADN’T THE HEART 33
44 HAS NOT THE MEANS 33
45 HAD NOT THE RIGHT 32
46 HAVE NOT THE KNOWLEDGE 32
47 HAS NOT THE LORD 31
48 HAVE NOT A WORD 31
49 HASN’T THE TIME 30
50 HAVE NOT THE LOVE 30
51 HATH NOT THE SON 29
52 HAVEN’T A CARE 28
53 HAVE NOT THE USE 28
54 HAS NOT THE STRENGTH 27
55 HAS NOT A LOT 26
56 HAVE NOT THE ABILITY 26
57 HAVE NOT THE HEART 26
58 HAVEN’T A LEG 25
59 HASN’T THE GOVERNMENT 25
60 HAD NOT A MOMENT 25
61 ‘VE NOT A CLUE 25
62 HAVE NOT A RIGHT 24
63 HAVE NOT THE FAITH 24
64 HADN’T A CHANCE 23
65 HAD NOT THE OPPORTUNITY 23
66 HAD NOT THE MONEY 23
67 HAVE NOT A MOMENT 23
68 HAVEN’T A NOTION 22
69 HAS NOT THE CHURCH 22
70 HAD NOT A PENNY 22
71 HAVEN’T THE SPACE 22
72 HAVE NOT THE OPPORTUNITY 22
73 HAVE NOT THE PATIENCE 22
74 HADN’T A HOPE 21
75 HASN’T A CARE 21
76 HAVE NOT THE MONEY 21
77 HAS NOT THE COURAGE 20
78 HATH NOT THE POTTER 20
79 HATH NOT THE FATHER 20
80 HAS NOT A THING 20
81 HAVEN’T THE ENERGY 20
82 HASN’T A HOPE 19
83 HAVE NOT A CARE 19
84 HAVE NOT AN IDEA 19
85 ‘VE NOT THE TIME 18
86 HADN’T THE COURAGE 18
87 HATH NOT THE SPIRIT 18
88 HAS NOT THE SON 18
89 HAVEN’T A LOT 18
90 HAVEN’T A THING 17
91 HAS NOT A CHANCE 17
92 HAD NOT THE PATIENCE 17
93 HAVEN’T THE GUTS 17
94 HAD NOT A RIGHT 16
95 HAD NOT A CHANCE 16
96 HADN’T THE STRENGTH 16
97 HASN’T THE RIGHT 16
98 HAVEN’T THE COURAGE 16
99 HAVEN’T A PROBLEM 16
100 HAVEN’T A PRAYER 16
_VH _XX _A _JJT
1 HAVE NOT THE SLIGHTEST 439
2 HAVEN’T THE SLIGHTEST 346
3 HAD NOT THE SLIGHTEST 261
4 HAVEN’T THE FAINTEST 232
5 HAVEN’T THE FOGGIEST 224
6 HAS NOT THE SLIGHTEST 144
7 HAD NOT THE FAINTEST 65
8 HAVE NOT THE FAINTEST 62
9 HADN’T THE SLIGHTEST 62
10 HASN’T THE SLIGHTEST 54
11 HADN’T THE FAINTEST 54
12 HAVE NOT THE SMALLEST 52
13 HAD NOT THE SMALLEST 35
14 HADN’T THE FOGGIEST 26
15 HAVE NOT THE FOGGIEST 25
16 HAS NOT THE FAINTEST 25
17 HAD NOT THE REMOTEST 24
18 HAVEN’T THE VAGUEST 23
19 HASN’T THE FOGGIEST 22
20 HASN’T THE FAINTEST 19