diff --git a/scipy/sparse/linalg/_isolve/tests/test_iterative.py b/scipy/sparse/linalg/_isolve/tests/test_iterative.py
index a782a8d27c2..48fc16b7a00 100644
--- a/scipy/sparse/linalg/_isolve/tests/test_iterative.py
+++ b/scipy/sparse/linalg/_isolve/tests/test_iterative.py
@@ -409,7 +409,9 @@ def test_atol(solver):
         residual = A.dot(x) - b
         err = np.linalg.norm(residual)
         atol2 = tol * b_norm
-        assert_(err <= max(atol, atol2))
+        # Added 1.00025 fudge factor because of `err` exceeding `atol` just
+        # very slightly on s390x (see gh-17839)
+        assert_(err <= 1.00025 * max(atol, atol2))
 
 
 @pytest.mark.parametrize("solver", [cg, cgs, bicg, bicgstab, gmres, qmr, minres, lgmres, gcrotmk, tfqmr])
@@ -452,8 +454,10 @@ def test_zero_rhs(solver):
                                                 and sys.version_info[1] == 9,
                                                 reason="gh-13019")),
     qmr,
-    pytest.param(lgmres, marks=pytest.mark.xfail(platform.machine() == 'ppc64le',
-                                                 reason="fails on ppc64le")),
+    pytest.param(lgmres, marks=pytest.mark.xfail(
+        platform.machine() not in ['x86_64' 'x86', 'aarch64', 'arm64'],
+        reason="fails on at least ppc64le, ppc64 and riscv64, see gh-17839")
+    ),
     pytest.param(cgs, marks=pytest.mark.xfail),
     pytest.param(bicg, marks=pytest.mark.xfail),
     pytest.param(bicgstab, marks=pytest.mark.xfail),
diff --git a/scipy/special/tests/test_orthogonal.py b/scipy/special/tests/test_orthogonal.py
index a325cb1a9ef..408b03bde2e 100644
--- a/scipy/special/tests/test_orthogonal.py
+++ b/scipy/special/tests/test_orthogonal.py
@@ -546,7 +546,7 @@ def test_roots_gegenbauer():
     vgq(rootf(170), evalf(170), weightf(170), -1., 1., 5, atol=1e-13)
     vgq(rootf(170), evalf(170), weightf(170), -1., 1., 25, atol=1e-12)
     vgq(rootf(170), evalf(170), weightf(170), -1., 1., 100, atol=1e-11)
-    vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 5, atol=1e-13)
+    vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 5, atol=1.25e-13)
     vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 25, atol=1e-12)
     vgq(rootf(170.5), evalf(170.5), weightf(170.5), -1., 1., 100, atol=1e-11)
 
diff --git a/scipy/stats/tests/test_distributions.py b/scipy/stats/tests/test_distributions.py
index eceeafdffe0..139b0537174 100644
--- a/scipy/stats/tests/test_distributions.py
+++ b/scipy/stats/tests/test_distributions.py
@@ -4254,7 +4254,7 @@ def test_pdf_nolan_samples(
             ],
             # for small alpha very slightly reduced accuracy
             [
-                'piecewise', 5e-11, lambda r: (
+                'piecewise', 2.5e-10, lambda r: (
                     np.isin(r['pct'], pct_range) &
                     np.isin(r['alpha'], alpha_range) &
                     np.isin(r['beta'], beta_range) &
@@ -4358,7 +4358,7 @@ def test_cdf_nolan_samples(
         tests = [
             # piecewise generally good accuracy
             [
-                'piecewise', 1e-12, lambda r: (
+                'piecewise', 2e-12, lambda r: (
                     np.isin(r['pct'], pct_range) &
                     np.isin(r['alpha'], alpha_range) &
                     np.isin(r['beta'], beta_range) &
@@ -4480,6 +4480,14 @@ def test_location_scale(
     ):
         """Tests for pdf and cdf where loc, scale are different from 0, 1
         """
+
+        uname = platform.uname()
+        is_linux_32 = uname.system == 'Linux' and "32bit" in platform.architecture()[0]
+        # Test seems to be unstable (see gh-17839 for a bug report on Debian
+        # i386), so skip it.
+        if is_linux_32 and case == 'pdf':
+            pytest.skip("Test unstable on some platforms; see gh-17839, 17859")
+
         data = nolan_loc_scale_sample_data
         # We only test against piecewise as location/scale transforms
         # are same for other methods.
diff --git a/scipy/stats/tests/test_mstats_basic.py b/scipy/stats/tests/test_mstats_basic.py
index cd53c7b7914..76cbfd8c253 100644
--- a/scipy/stats/tests/test_mstats_basic.py
+++ b/scipy/stats/tests/test_mstats_basic.py
@@ -1768,8 +1768,8 @@ def test_skewtest_2D_WithMask(self):
                 r = stats.skewtest(x)
                 rm = stats.mstats.skewtest(xm)
 
-                assert_allclose(r[0][0], rm[0][0], rtol=2e-15)
-                assert_allclose(r[0][1], rm[0][1], rtol=1e-15)
+                assert_allclose(r[0][0], rm[0][0], rtol=1e-14)
+                assert_allclose(r[0][1], rm[0][1], rtol=1e-14)
 
     def test_normaltest(self):
         with np.errstate(over='raise'), suppress_warnings() as sup: