Anywhere I Lay My Head contains four songs written by Tom Waits, six songs written by Waits and his wife Kathleen Brennan, and one original composition, "Song for Jo". Upon release, the album received mixed reviews from critics, and saw moderate commercial success. "Falling Down" was released as the album's lead single.
Anywhere I Lay My Head received mixed reviews from music critics. At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream critics, the album received an average score of 58, based on 35 reviews, indicating "mixed or average reviews". Priya Elan of the NME called the album "brilliant" and wrote that "just like Lou Reed with Nico and Serge Gainsbourg with Brigitte Bardot, Sitek has effortlessly translated Johansson's magnetism on to record", while comparing her "deep" voice to "latter-day Ronnie Spector's street-savvy tone".The Observer's Barney Hoskyns commented that Johansson's "blankly androgynous alto timbre is nothing special, but that barely matters", praising the album as "a bravely eccentric selection and a captivating homage to a singular writer".The Guardian's Dorian Lynskey characterized Johansson's voice as "a supple, languid instrument offering hints of Nico, Kim Deal and Martina Topley-Bird" and stated, "You might wish there was more from Waits' 70s barfly period [...] but it's a measure of this album's surprising allure that you're left wanting more."
Allmusic reviewer Stephen Thomas Erlewine found Johansson to be "surprisingly deep and brittle as a singer", concluding that the album "doesn't quite work, but it can't quite be dismissed, either: unlike so many actor-turned-singer records, there's not a hint of vanity to this project and it's hard not to marvel at its ambition even as it fails." Matt Fink of Paste magazine stated that Johansson's "willingness to allow the arrangements to transform Waits' creaky intimacy into wide-eyed atmosphere ultimately results in the rare covers album that actually has its own identity." In another review for Paste, however, Amanda Petrusich felt that Johansson's singing "feels strangely sluggish and plodding", writing, "These songs were already so impeccably performed that Johansson didn't have very many new places to take them, and although her effort and nerve are commendable, 'not as terrible as you thought it would be' just isn't the same thing as good."
The Village Voice's Stacey Anderson remarked that Johansson is "a monotonous singer, with a garrulous flat alto that swoops uneasily to sub-Nico baritone [...] But thankfully, Sitek recognizes her limitations and wisely pairs her with Waits's most frankly plaintive lyrics until the emotion drips through". Mikael Wood of Spin magazine wrote, "Beyond the fact that her voice is deep enough for her to front Crash Test Dummies, there's nothing particularly compelling about Scarlett Johansson's singing", adding that "her vocals are buried deep beneath [...] Dave Sitek's mountain of reverbed space-gospel noise." Wood nevertheless noted the album is "[n]ot your typical Hollywood vanity project". Chris Willman of Entertainment Weekly opined, "In burying Johansson's vocals so deeply in the druggy ambiance, producer David Andrew Sitek [...] means well but ends up obscuring Waits' great tunes."Rolling Stone's Will Hermes critiqued that "Johansson's voice is unremarkable and her pitch sometimes unsteady", dubbing her "a faintly goth Marilyn Monroe lost in a sonic fog". Hermes complimented the tracks "I Wish I Was in New Orleans", "Fannin Street", and "Song for Jo", but claimed that "the synth-pop version of 'I Don't Wanna Grow Up,' famously covered by the Ramones, makes you wish Joey was still around to take the mic."
Stephen M. Deusner of Pitchfork Media viewed the album as "a Brooklyn update on vintage 4AD bands like This Mortal Coil or Cocteau Twins", but ultimately noted that "[t]he only thing we've learned about her is that she really, really likes Tom Waits. That's more than enough to avoid catastrophe, but not quite enough to make Anywhere I Lay My Head much more than a curio."Slant Magazine's Dave Hughes expressed that Johansson is "neither a particularly interesting nor a particularly skillful singer, and she spends much of the record locked into a sub-Nico hum that's quite a bit less charismatic than her husky line readings might suggest."Drowned in Sound's Alex Denney concluded, "Perversely given the record's comprehensive musical overhaul it's perhaps a surfeit of respect for the source material that proves Anywhere...'s undoing; for all its undoubted accomplishments there's a lingering suspicion that this is too safe, too respectable a record to do justice to an artist who remains forever mid-topple from the bar stool in the popular consciousness."
Anywhere I Lay My Head debuted at number 126 on the Billboard 200, selling 5,100 copies in its first week. The album fared better in Europe, reaching number fifteen in Switzerland, number twenty-five in Austria, number twenty-six in France, number twenty-seven in Sweden, and number thirty in Belgium and Germany. By February 2010, the album had sold about 70,000 copies worldwide.