Would they carry very narrow rapier in battlefield or hand to hand combat in ship?And what sword would their Navy soldier use?

One distinctly Portuguese form of hilt seems to be the "crab" or "testicles" style.
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Since 1400 they had a slightly modified iberian sword called "Colhona", or "Crab Sword" of "Black Carracks Sword", known for being as much as a cut-and-thrust sword with protective rings and a blackened blade (intended for use in salted water). But I think these fell out of favor at later decades stages of 16th century. Otherwise all other swords, specially knights and fidalgos' swords, would be similar to what the spaniards or the english had at that time.

By the way, do you have Osprey's "The Portuguese in the Age of Discovery"? It's the best guide I have ever find about portuguese military so far. I have the ebook and I can send you by email

I am getting into 15th c living history and interested in getting a reproduction.

For the later 15th century, You might be interested in this book chapter which gives a really good discussion of the portugese arms and armor depicted in the Pastrana tapestries:

https://www.academia.edu/8267811/Afonso_the_African_and_his_Army_The_Pastrana_Tapestries_as_a_Visual_Encyclopedia_for_the_Study_of_Arms_and_Armor

Here are images of the various tapestries themselves from wikimedia:

https://commons.wikimedia.org/wiki/Category:Pastrana_tapestries