diff --git a/jax/_src/lax/svd.py b/jax/_src/lax/svd.py
index cf0ffa562ef7..1657ea4c9d63 100644
--- a/jax/_src/lax/svd.py
+++ b/jax/_src/lax/svd.py
@@ -42,21 +42,46 @@
 from jax._src import core
 
 
-@functools.partial(jax.jit, static_argnums=(1, 2))
-def _zero_svd(a: Any,
-              full_matrices: bool,
-              compute_uv: bool = True) -> Union[Any, Sequence[Any]]:
+@functools.partial(jax.jit, static_argnums=(2, 3))
+def _constant_svd(
+    a: Any, return_nan: bool, full_matrices: bool, compute_uv: bool = True
+) -> Union[Any, Sequence[Any]]:
   """SVD on matrix of all zeros."""
   m, n = a.shape
   k = min(m, n)
-  s = jnp.zeros(shape=(k,), dtype=a.real.dtype)
+  s = jnp.where(
+      return_nan,
+      jnp.full(shape=(k,), fill_value=jnp.nan, dtype=a.real.dtype),
+      jnp.zeros(shape=(k,), dtype=a.real.dtype),
+  )
   if compute_uv:
+    fill_value = (
+        jnp.nan + 1j * jnp.nan
+        if jnp.issubdtype(a.dtype, jnp.complexfloating)
+        else jnp.nan
+    )
     if full_matrices:
-      u = jnp.eye(m, m, dtype=a.dtype)
-      vh = jnp.eye(n, n, dtype=a.dtype)
+      u = jnp.where(
+          return_nan,
+          jnp.full((m, m), fill_value, dtype=a.dtype),
+          jnp.eye(m, m, dtype=a.dtype),
+      )
+      vh = jnp.where(
+          return_nan,
+          jnp.full((n, n), fill_value, dtype=a.dtype),
+          jnp.eye(n, n, dtype=a.dtype),
+      )
     else:
-      u = jnp.eye(m, k, dtype=a.dtype)
-      vh = jnp.eye(k, n, dtype=a.dtype)
+      u = jnp.where(
+          return_nan,
+          jnp.full((m, k), fill_value, dtype=a.dtype),
+          jnp.eye(m, k, dtype=a.dtype),
+      )
+      vh = jnp.where(
+          return_nan,
+          jnp.full((k, n), fill_value, dtype=a.dtype),
+          jnp.eye(k, n, dtype=a.dtype),
+      )
     return (u, s, vh)
   else:
     return s
@@ -86,6 +111,9 @@ def _svd_tall_and_square_input(
 
   # TODO: Uses `eigvals_only=True` if `compute_uv=False`.
   v, s = lax.linalg.eigh(h)
+  # Singular values are non-negative by definition. But eigh could return small
+  # negative values, so we clamp them to zero.
+  s = jnp.maximum(s, 0.0)
 
   # Flips the singular values in descending order.
   s_out = jnp.flip(s)
@@ -228,11 +256,22 @@ def svd(a: Any,
   # X_{k+1} = X_k(a_k I + b_k {X_k}^H X_k)(I + c_k {X_k}^H X_k)^{−1} and
   # X_0 = A/alpha, where alpha = ||A||_2, the triplet (a_k, b_k, c_k) are
   # weighting parameters, and X_k denotes the k^{th} iterate.
-  return jax.lax.cond(jnp.all(a == 0),
-                      functools.partial(_zero_svd, full_matrices=full_matrices,
-                                        compute_uv=compute_uv),
-                      functools.partial(_qdwh_svd, full_matrices=full_matrices,
-                                        compute_uv=compute_uv,
-                                        hermitian=hermitian,
-                                        max_iterations=max_iterations),
-                      operand=(a))
+  all_zero = jnp.all(a == 0)
+  non_finite = jnp.logical_not(jnp.all(jnp.isfinite(a)))
+  return lax.cond(
+      jnp.logical_or(all_zero, non_finite),
+      functools.partial(
+          _constant_svd,
+          return_nan=non_finite,
+          full_matrices=full_matrices,
+          compute_uv=compute_uv,
+      ),
+      functools.partial(
+          _qdwh_svd,
+          full_matrices=full_matrices,
+          compute_uv=compute_uv,
+          hermitian=hermitian,
+          max_iterations=max_iterations,
+      ),
+      operand=(a),
+  )
diff --git a/tests/svd_test.py b/tests/svd_test.py
index b11ed508b22a..abb1f4037194 100644
--- a/tests/svd_test.py
+++ b/tests/svd_test.py
@@ -173,21 +173,44 @@ def testSingularValues(self, m, n, log_cond, full_matrices):
         np.testing.assert_array_less(actual_diff, np.zeros_like(actual_diff))
 
   @jtu.sample_product(
-    [dict(m=m, n=n) for m, n in zip([2, 4, 8], [4, 4, 6])],
-    full_matrices=[True, False],
-    compute_uv=[True, False],
-    dtype=jtu.dtypes.floating + jtu.dtypes.complex,
+      [dict(m=m, n=n) for m, n in zip([2, 4, 8], [4, 4, 6])],
+      full_matrices=[True, False],
+      compute_uv=[True, False],
+      dtype=jtu.dtypes.floating + jtu.dtypes.complex,
   )
-  def testSvdOnZero(self, m, n, full_matrices, compute_uv, dtype):
-    """Tests SVD on matrix of all zeros."""
-    osp_fun = functools.partial(osp_linalg.svd, full_matrices=full_matrices,
-                                compute_uv=compute_uv)
-    lax_fun = functools.partial(svd.svd, full_matrices=full_matrices,
-                                compute_uv=compute_uv)
+  def testSvdAllZero(self, m, n, full_matrices, compute_uv, dtype):
+    """Tests SVD on matrix of all zeros, +/-infinity or NaN."""
+    osp_fun = functools.partial(
+        osp_linalg.svd, full_matrices=full_matrices, compute_uv=compute_uv
+    )
+    lax_fun = functools.partial(
+        svd.svd, full_matrices=full_matrices, compute_uv=compute_uv
+    )
     args_maker_svd = lambda: [jnp.zeros((m, n), dtype=dtype)]
     self._CheckAgainstNumpy(osp_fun, lax_fun, args_maker_svd)
     self._CompileAndCheck(lax_fun, args_maker_svd)
 
+  @jtu.sample_product(
+      [dict(m=m, n=n) for m, n in zip([2, 4, 8], [4, 4, 6])],
+      fill_value=[-np.inf, np.inf, np.nan],
+      full_matrices=[True, False],
+      compute_uv=[True, False],
+      dtype=jtu.dtypes.floating + jtu.dtypes.complex,
+  )
+  def testSvdNonFiniteValues(
+      self, m, n, fill_value, full_matrices, compute_uv, dtype
+  ):
+    """Tests SVD on matrix of all zeros, +/-infinity or NaN."""
+    lax_fun = functools.partial(
+        svd.svd, full_matrices=full_matrices, compute_uv=compute_uv
+    )
+    args_maker_svd = lambda: [
+        jnp.full((m, n), fill_value=fill_value, dtype=dtype)
+    ]
+    result = lax_fun(args_maker_svd()[0])
+    for r in result:
+      self.assertTrue(jnp.all(jnp.isnan(r)))
+    self._CompileAndCheck(lax_fun, args_maker_svd)
 
   @jtu.sample_product(
     [dict(m=m, n=n, r=r, c=c)