This is the result of a challenge from Karl Scherer, who already has a large results page posted. Below are some results I have found. Some of these are also on Karl's page, for solutions that I either found first, or found independently before seeing them on his solutions page.

The object of this challenge is to tile rectangles and squares with squares in a "nowhere-neat" way. Nowhere-neat means that no two squares share two vertices and an entire edge.

A secondary challenge is to tile rectangles and squares without same-size squares sharing any segment of edge. These may be called "no-touch" solutions.

Tilings on this page: 11x15, 16x16, 19x19, 20x20, 20x85, 22x40, 23x23 (*), 27x27 (*), 28x43, 28x48, 32x32 (*, two solutions), 35x35 (*, two solutions), 37x37 (*), 38x38 (*), 39x39, (two solutions), 40x40 (two solutions, one *), 45x45, 47x47, 53x53 (two solutions), 55x55, 56x56, 59x59 (two solutions), 60x60, 65x65, 68x68, 71x71, 75x75, 83x83, 98x98, 105x105

(*) Different than the solution(s) on Karl's page for this size.

* marks no-touch solutions:

11x15 *

16x16 *

19x19

20x20

20x85 *

22x40

23x23

27x27 *

28x43

28x48

32x32

32x32 *

35x35

35x35

37x37

38x38

39x39 *

39x39

40x40

40x40 *

45x45

47x47 *

53x53

53x53 *

55x55 *

56x56

59x59 *

59x59

60x60

65x65

68x68 *

71x71

75x75

83x83

98x98

105x105 *

11x15 *

16x16 *

19x19

20x20

20x85 *

22x40

23x23

27x27 *

28x43

28x48

32x32

32x32 *

35x35

35x35

37x37

38x38

39x39 *

39x39

40x40

40x40 *

45x45

47x47 *

53x53

53x53 *

55x55 *

56x56

59x59 *

59x59

60x60

65x65

68x68 *

71x71

75x75

83x83

98x98

105x105 *