Most western governments claim to have evidence of Iran military involvement in Yemen, including weapons shipments, that they keep classified.
However, here are some concrete elements of proof that have reached the world's medias. How decisive they are is up to you; Western governements', Saudi's and Iran's reactions about them have widely differed.
2013 : Ship Jihan I
Yemeni authorities have seized an Iranian boat full of weapons:
Yemeni authorities point to the “Jihan 1” as evidence of Iran’s support. The ship was seized by Yemen in 2013, smuggling weapons from Iran to local insurgents. The Yemeni official showed Reuters a breakdown of the cargo, which included Katyusha rockets M-122, heat-seeking surface-to-air missiles, RPG-7s, Iranian-made night vision goggles and “artillery systems that track land and navy targets 40km away”.
But of course:
Iran denied any connection with the arms found on Jihan 1.
2015 : UN 'secret' report
In April 2015, a report by UN expert was transfered to the Iran Sanction Committee but has leaked to the medias (notably French AFP) and been widely reported.
Studying evidence that includes afore-mentioned Jihan I, it suggests that weapons were delivered by sea from Iran to Yemen "at least since 2009":
Les informations recueillies "suggèrent que le cas du Jihan suit d'autres livraisons par mer au Yémen que l'on peut faire remonter au moins à 2009", indique le rapport que l'AFP a pu consulter.
The collected informations "suggest that the Jihan case follows other shipments by sea to Yemen that can be dated back at least to 2009", says the rapport AFP could consult. - my translation.
2017: new UN report
On dec 1, 2017, Reuters says it has had access to another report, dated nov 24, 2017:
Remnants of four ballistic missiles fired into Saudi Arabia by Yemen’s Houthi rebels this year appear to have been designed and manufactured by Riyadh’s regional rival Iran, a confidential report by United Nations sanctions monitors said.
The existence of the report was confirmed on dec 13, 2017, when the UN Secretary General Antonio
Guterres said in a report to the security council that the United Nations was investigating Iran’s possible transfer of ballistic missiles to Houthi Shia rebels in Yemen that may have been used in launches aimed at Saudi Arabia in July and November.
2017 : Remnants of ballistic missiles shown in Bolling Air Force base in Washington.
On Nov,14,2017, Nicky Haley presented short-range ballistic missiles that are said to have been built in Iran, delivered to Houthis and fired in Yemen.
Standing in front of segments of two missiles, which US officials say were fired recently by Houthi forces at Saudi Arabia, Haley said: “As you know we do not often declassify this time of military equipment recovered from these attacks but today we are taking an extraordinary step of presenting it here in an opening setting.”
“In this warehouse is concrete evidence of illegal Iranian weapons proliferation gathered by direct military attacks on our partners in the regime,” she added, saying that representatives from other countries had been invited to inspect the evidence at Bolling.
Sure, US officials displaying proofs of enemy weapons is taken with a bit of salt at least since Colin Powell... Guess what Iran's comment has been ?
The Iranian spokesman at the UN, Alireza Miryousefi, said the evidence was fake.
He said: “We categorically reject it as unfounded and, at the same time, irresponsible, provocative and destructive. This purported evidence, put on public display today, is as much fabricated as the one presented on some other occasions earlier.”