Controversy surrounds the inclusion of nutritional supplements in mental health treatments, with much research proving inconclusive. A major new review now explains which supplements have shown the most promise for specific mental health conditions.

...However, they note, a few supplements did help improve specific symptoms of particular disorders "under certain conditions" and when the person took them alongside other treatments.
The meta-analyses that the researchers assessed indicated that omega-3 supplements could help relieve symptoms of major depression in people who also take antidepressants.

...For mood disorders and schizophrenia, N-acetylcysteine — an amino acid — seemed to help when individuals took it alongside their regular treatments.

Various dosages of folate-based supplements seemed to help with managing symptoms of depression and schizophrenia — though folic acid did not have this effect.

...Yet when it comes to vitamins and minerals, the researchers found no compelling evidence to suggest that these could help manage any symptoms related to mental health.

More: Major review asks which supplements really aid mental health

Study - full text: The efficacy and safety of nutrient supplements in the treatment of mental disorders
Magnesium supplements should also be considered, imho, as there is some evidence it has a positive impact on anxiety and depression, plus it may help ease some antidepressant and antipsychotic side-effects such as restless legs.
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Of the two main fatty acids in Omega-3/fish oil, eicosapentaenoic acid (EPA) seems to be more important than docosahexaenoic acid (DHA) for easing anxiety and depression so if the cost is about the same choose the brand with the highest EPA-DHA ratio (Liao Y, 2019; Mocking RJ, 2016; Grosso G, 2014; Samieri C, 2012; Sublette ME, 2011; Martins JG, 2009).

Caution: Both fish oil and serotonergic antidepressants are mild anticoagulants which can be beneficial as they may reduce heart attack and ischaemic stroke risk (but may slightly increase that of the much less common hemorrhagic stroke), however, there is a risk of bleeding when the supplement is taken at high doses together with SSRIs/SNRIs/some TCAs. Fish oil at the recommended dose is usually safe, but lower the dose if you begin to bruise more easily, or develop nose, or gum bleeding. Also avoid taking aspirin and other NSAID painkillers such as ibuprofen and naproxen, which also 'thin' the blood. Paracetamol, aka acetaminophen, is the preferred painkiller.

N-Acetyl-Cysteine (NAC) is generally a safe supplement with few side-effects. It is mostly used medically to protect the liver in cases of paracetamol, aka acetaminophen, overdose and has other medical uses. NAC has been shown to be particularly effective in reducing OCD intrusive and compulsive thought patterns and also in boosting the effectiveness of antidepressants for OCD and other OCD spectrum disorders [1].

Doses of 1,800-3,000mg/day taken in 3 divided doses seems to work (start with one dose per day of around 500-600mg and increase by the same amount every 2-3 days to avoid triggering diarrhoea). NAC is best taken on an empty stomach about an hour before meals.

Caution: a study [2] found NAC (and Vitamin E) may increase the risk of lung cancer in smokers, or those with some specific lung diseases (note, the study found it doesn't cause lung cancer, only maybe reduces the ability of the immune system to detect and destroy existing cancerous cells). This finding has been the subject of much debate and considerable scepticism because there are many other studies showing NAC protects against lung cancer so it is probably not the final word on the subject. FWIW, I'm a lung cancer survivor and have take NAC for over 12 years because it is the best thing I've found for reducing fluid build up in my remaining lung. As of 6 months ago I was cancer free.

Seek the advice of your GP, or pharmacist before taking NAC. While NAC can be beneficial for asthmatics with fluid build up in lungs, it may cause shortness of breath in some, so caution is needed if you have asthma. It is contraindicated for those taking nitroglycerin, vitamin K and some blood pressure meds.

[1]
Couto JP, Moreira R. (2018)
Oral N-acetylcysteine in the treatment of obsessive-compulsive disorder: A systematic review of the clinical evidence.
Prog Neuropsychopharmacol Biol Psychiatry. Aug 30;86:245-254 (Abstract)

di Michele F, Siracusano A, Talamo A, Niolu C. (2018)
N-Acetyl Cysteine and Vitamin D Supplementation in Treatment Resistant Obsessive-compulsive Disorder Patients: A General Review.
Curr Pharm Des. 24(17):1832-1838 (Abstract)

Ghanizadeh A1,2, Mohammadi MR3,4, Bahraini S, etal. (2017)
Efficacy of N-Acetylcysteine Augmentation on Obsessive Compulsive Disorder: A Multicenter Randomized Double Blind Placebo Controlled Clinical Trial.
Iran J Psychiatry. Apr;12(2):134-141 (Abstract | Full text)

Oliver G, Dean O, Camfield D4, Blair-West S, et al (2015)
N-acetyl cysteine in the treatment of obsessive compulsive and related disorders: a systematic review.
Clin Psychopharmacol Neurosci. Apr 30;13(1):12-24 (Abstract | Full text - PDF)

[2]
Sayin VI1, Ibrahim MX, Larsson E, et al. (2014)
Antioxidants accelerate lung cancer progression in mice.
Sci Transl Med. 2014 Jan 29;6(221):221ra15 (Abstract | Full text - PDF)