diff --git a/pkg/textparse/openmetricsparse_test.go b/pkg/textparse/openmetricsparse_test.go
index 2da3013f8..de672e3c2 100644
--- a/pkg/textparse/openmetricsparse_test.go
+++ b/pkg/textparse/openmetricsparse_test.go
@@ -498,6 +498,10 @@ func TestOpenMetricsParseErrors(t *testing.T) {
 			input: `custom_metric_total 1 # {aa=\"\xff\"} 9.0`,
 			err:   "expected label value, got \"INVALID\"",
 		},
+		{
+			input: `{b="c",} 1`,
+			err:   `"INVALID" "{" is not a valid start token`,
+		},
 	}
 
 	for i, c := range cases {
diff --git a/pkg/textparse/promparse_test.go b/pkg/textparse/promparse_test.go
index 9b62079d5..28761ed13 100644
--- a/pkg/textparse/promparse_test.go
+++ b/pkg/textparse/promparse_test.go
@@ -265,6 +265,10 @@ func TestPromParseErrors(t *testing.T) {
 			input: "foo 0 1_2\n",
 			err:   "expected next entry after timestamp, got \"MNAME\"",
 		},
+		{
+			input: `{a="ok"} 1`,
+			err:   `"INVALID" is not a valid start token`,
+		},
 	}
 
 	for i, c := range cases {
diff --git a/scrape/scrape.go b/scrape/scrape.go
index 99bc85fbb..c22f626b0 100644
--- a/scrape/scrape.go
+++ b/scrape/scrape.go
@@ -47,6 +47,8 @@ import (
 	"github.com/prometheus/prometheus/storage"
 )
 
+var errNameLabelMandatory = fmt.Errorf("missing metric name (%s label)", labels.MetricName)
+
 var (
 	targetIntervalLength = prometheus.NewSummaryVec(
 		prometheus.SummaryOpts{
@@ -1154,6 +1156,11 @@ loop:
 				continue
 			}
 
+			if !lset.Has(labels.MetricName) {
+				err = errNameLabelMandatory
+				break loop
+			}
+
 			var ref uint64
 			ref, err = app.Add(lset, t, v)
 			switch err {
diff --git a/scrape/scrape_test.go b/scrape/scrape_test.go
index 9c9fe9e45..d05b3c400 100644
--- a/scrape/scrape_test.go
+++ b/scrape/scrape_test.go
@@ -1587,6 +1587,41 @@ func TestScrapeLoopDiscardDuplicateLabels(t *testing.T) {
 	testutil.Equals(t, false, series.Next(), "more than one series found in tsdb")
 }
 
+func TestScrapeLoopDiscardUnnamedMetrics(t *testing.T) {
+	s := teststorage.New(t)
+	defer s.Close()
+
+	app := s.Appender()
+
+	ctx, cancel := context.WithCancel(context.Background())
+	sl := newScrapeLoop(ctx,
+		&testScraper{},
+		nil, nil,
+		func(l labels.Labels) labels.Labels {
+			if l.Has("drop") {
+				return labels.Labels{}
+			}
+			return l
+		},
+		nopMutator,
+		func() storage.Appender { return app },
+		nil,
+		0,
+		true,
+	)
+	defer cancel()
+
+	_, _, _, err := sl.append([]byte("nok 1\nnok2{drop=\"drop\"} 1\n"), "", time.Time{})
+	testutil.NotOk(t, err)
+	testutil.Equals(t, errNameLabelMandatory, err)
+
+	q, err := s.Querier(ctx, time.Time{}.UnixNano(), 0)
+	testutil.Ok(t, err)
+	series, _, err := q.Select(&storage.SelectParams{}, labels.MustNewMatcher(labels.MatchRegexp, "__name__", ".*"))
+	testutil.Ok(t, err)
+	testutil.Equals(t, false, series.Next(), "series found in tsdb")
+}
+
 func TestReusableConfig(t *testing.T) {
 	variants := []*config.ScrapeConfig{
 		&config.ScrapeConfig{
diff --git a/tsdb/db_test.go b/tsdb/db_test.go
index 52cccfabd..2493ffea6 100644
--- a/tsdb/db_test.go
+++ b/tsdb/db_test.go
@@ -357,12 +357,12 @@ func TestAmendDatapointCausesError(t *testing.T) {
 	}()
 
 	app := db.Appender()
-	_, err := app.Add(labels.Labels{}, 0, 0)
+	_, err := app.Add(labels.Labels{{Name: "a", Value: "b"}}, 0, 0)
 	testutil.Ok(t, err)
 	testutil.Ok(t, app.Commit())
 
 	app = db.Appender()
-	_, err = app.Add(labels.Labels{}, 0, 1)
+	_, err = app.Add(labels.Labels{{Name: "a", Value: "b"}}, 0, 1)
 	testutil.Equals(t, ErrAmendSample, err)
 	testutil.Ok(t, app.Rollback())
 }
@@ -375,12 +375,12 @@ func TestDuplicateNaNDatapointNoAmendError(t *testing.T) {
 	}()
 
 	app := db.Appender()
-	_, err := app.Add(labels.Labels{}, 0, math.NaN())
+	_, err := app.Add(labels.Labels{{Name: "a", Value: "b"}}, 0, math.NaN())
 	testutil.Ok(t, err)
 	testutil.Ok(t, app.Commit())
 
 	app = db.Appender()
-	_, err = app.Add(labels.Labels{}, 0, math.NaN())
+	_, err = app.Add(labels.Labels{{Name: "a", Value: "b"}}, 0, math.NaN())
 	testutil.Ok(t, err)
 }
 
@@ -392,15 +392,28 @@ func TestNonDuplicateNaNDatapointsCausesAmendError(t *testing.T) {
 	}()
 
 	app := db.Appender()
-	_, err := app.Add(labels.Labels{}, 0, math.Float64frombits(0x7ff0000000000001))
+	_, err := app.Add(labels.Labels{{Name: "a", Value: "b"}}, 0, math.Float64frombits(0x7ff0000000000001))
 	testutil.Ok(t, err)
 	testutil.Ok(t, app.Commit())
 
 	app = db.Appender()
-	_, err = app.Add(labels.Labels{}, 0, math.Float64frombits(0x7ff0000000000002))
+	_, err = app.Add(labels.Labels{{Name: "a", Value: "b"}}, 0, math.Float64frombits(0x7ff0000000000002))
 	testutil.Equals(t, ErrAmendSample, err)
 }
 
+func TestEmptyLabelsetCausesError(t *testing.T) {
+	db, closeFn := openTestDB(t, nil, nil)
+	defer func() {
+		testutil.Ok(t, db.Close())
+		closeFn()
+	}()
+
+	app := db.Appender()
+	_, err := app.Add(labels.Labels{}, 0, 0)
+	testutil.NotOk(t, err)
+	testutil.Equals(t, "empty labelset: invalid sample", err.Error())
+}
+
 func TestSkippingInvalidValuesInSameTxn(t *testing.T) {
 	db, closeFn := openTestDB(t, nil, nil)
 	defer func() {
diff --git a/tsdb/head.go b/tsdb/head.go
index 0fde4c845..e1eafcd30 100644
--- a/tsdb/head.go
+++ b/tsdb/head.go
@@ -957,6 +957,10 @@ func (a *headAppender) Add(lset labels.Labels, t int64, v float64) (uint64, erro
 	// Ensure no empty labels have gotten through.
 	lset = lset.WithoutEmpty()
 
+	if len(lset) == 0 {
+		return 0, errors.Wrap(ErrInvalidSample, "empty labelset")
+	}
+
 	if l, dup := lset.HasDuplicateLabelNames(); dup {
 		return 0, errors.Wrap(ErrInvalidSample, fmt.Sprintf(`label name "%s" is not unique`, l))
 	}