Proof required of cigarette or weed smoking?

Don't prove it, just serve them with Section 21 Notice without giving any reason.

Simon is right, provided the tenancy's out of its fixed term (or fixed term is ending within the next two months) and you've fully complied with legislation (come back if you need to clarify this) you can serve s.21 - no reason is needed.
If the tenancy is still in fixed term, there is nothing you can effectively do other than send a warning letter, but that's not going to stop something like that and may just escalate the problem (depending on your tenants' personalities). Where cannabis smoking can be a real problem is when it's done in common areas of blocks of flats or gets into adjoining properties as the smell is strong (though when I've experienced it the perpetrators didn't realise that was happening and stopped when it was pointed out to them).
If you're nowhere near the end of the fixed term, you will only be able to evict if the tenants run up more than 2 months of rent arrears.
You can make a deduction from the deposit at the end of the tenancy for the additional cleaning and decorating.

Can I ask why you are so bother about this? Personally I am a non smoker and have never smoked weed, however if I had a tenant who did but paid their rent on time, kept the house tidy and didn't cause a nuisance to the neighbours and looked like they were going to stay long term say 10 years or so I would over look it as long as the tenant was good in any every other way. I don't want voids in my properties, there is worse things a tenant could be doing ie not paying me and I try to think that we all lead our lives differently.

Although it may be a breach of a term/condition of the tenancy agreement, if there are no other issues then why make such a big fuss about it? Is it causing a nuisance to any other residents, or neighbours? By all means send a letter reminding them that use of drugs in the house is a breach of tenancy, but you have to be very careful not to accuse them of something that you have no actual evidence of, so the wording would need to be more of a reminder of the tenancy terms, rather than an accusation and warning. Even if you could prove that they were smoking cigarettes or cannabis in the house, it would not give you certain grounds for evicting them, and you could waste a lot of money taking them to court to achieve nothing. And like Janet says, why create the voids and loss of rent? If you are concerned about the cost of cleaning or re-decorating, then you can deduct it from their deposit, or just put the rent up at the next rent review to cover the likely cleaning/redecorating costs.
If determined to evict for such a minor breach of your tenancy agreement, then like Mandy says, you would simply issue the s21 Notice at the correct time so as to bring the tenancy to an end at the end of the fixed term. But beware, this will cost you a lot of money and could potentially cause you a lot more problems.

I think some of the comments are a bit complacent or is this so common that complacency is the only thing possible? I don't know without extensive investigations the law, but is there any legal implications for the Landlord if their tenants use or store drugs or have their tenants dealing in their properties?

A long time ago I had a tenant growing Marijuana in his room. As he was a student and due to leave within six months I said nothing, (I didn't want to harm this students as I didn't think this should be illegal which it was at that time), although I ensured everything was gone before I repossessed the property six months later. It was a great worry.

Of course there are rules about smoking in an HMO in common places, besides a landlord can set his or her own rules how his property should be treated, I am a smoker, I smoke in my own house, but rules are rules, in an HMO no one is allowed to smoke in common place, not even I, so when I told one of my tenants that smoking is strictly prohibited indoors, but fully allowed outside in rear or front gardens, one tenant kept on ignoring this rule, so one day I turned up at the property and he was caught smoking a cigarette red handed, and he smiled and continued to puff smoke, I gently told him to stub it out or go out, he refused, and continued to smile and smoke, I saw this as an intimidation, and normally as a landlord we do get subject to a load of intimidation as the tenants know the housing law that aids their ill behavior, but this time it was his last time he could puff any cigarette in my house I rent as an HMO, I stepped forward and took his cigarette out of his mouth and stubbed it in the sink as he was standing next to kitchen sink, he got very annoyed about it but I reminded him that rules are rules, follow them or get out and find another place. I have not since caught him smoking indoors even when it has been raining outside, he is welcome to use the garden shed I have told him.. His wife also smokes, but whenever I came to check my premises and she saw me, she would immidiately stub her cigarette out of respect even though she is not allowed to smoke in common parts, but I have extended this rule to cover the entire building , so this clause is in my contract, just as public places and buildings impose No Smoking Rules on smokers. No smoking permitted in bedrooms either.

Reply to the comment left by Yvonne Francis at 06/02/2018 - 11:07
It is not so much a matter of complacency, it is a matter of facing reality. The police do not care about people smoking cannabis (dealing and/or growing it are much more likely to be acted upon), and the courts are unlikely to grant a landlord outright possession even if the landlord could prove that the tenant had smoked cannabis in the property, so what can a landlord actually do???? Yes, they can write it in to their tenancy agreements, yes they can send a warning letter, yes they can serve a s21 Notice (eventually), but in real terms there is very little else that the law will allow a landlord to do. The law does not give a landlord the power to take any effective action to prevent such behaviour, indeed if the landlord did try to take action he/she could find themselves facing an accusation of harassment, and as we know, the law gives all rights to the tenant and no protection to the landlord.
Please please please prove me wrong and point me to some law that gives landlords the power to easily evict tenants found in possession of drugs (preferably without the £300+ court costs, and 4+ months of waiting for notice periods to expire, waiting for court hearings, and waiting for bailiffs, etc).

There can be all sorts of well meaning advice but at the end of the day it's 'you" that has to face the music! You don't know how far things have gone but you could check the loft! I've been there, it's not worth the stress, issue s21 for piece of mind or pay the price as I did. In short, "cover yourself".

Reply to the comment left by Robert Mellors at 06/02/2018 - 11:27
I don't think you have taken my post in the way I intended. I realise the police won't take action, that's why I said 'complacency is the is the only thing possible'. My point, although I don't know much about the law on this but hope someone else will, is that a Landlord could be responsible for any drug activity from smoking to supply carried out by tenants on the Landlords property. After all I'm responsible and can be heavily fined for the cleanliness of my tenants cookers despite having them professionally cleaned at the beginning of their tenancy.

Reply to the comment left by Robert Mellors at 06/02/2018 - 11:27
I can assure you the police do care, I had the same issue brought to my attention by the police due to complaints from the neighbours. It was my choice, deal with the problem tenant or face a closure order by the council and police with powers under the anti-social behaviour, crime and policing act 2014, tenants would be moved out and the property would become vacant resulting in loss of rent. A Closure Order is a power under the Anti-Social Behaviour, Crime and Policing Act 2014, which we have used before in the district, which could be open for consideration if we cannot resolve the issues at the property and whoever is responsible within does not desist. To give you an update in terms of the law around it, in case you are not familiar with the act, following an initial notice being issued by a senior police officer, which prevents anyone other than habitual residents entering the property, it must then be considered by the Magistrates Court, where it can be applied in a number of ways, with the most restrictive option preventing anyone including the landlord and tenants from entering for a period of initially not more than 3 months.Therefore it was in my hands to deal with the culprit. They are no longer staying in my property.